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The Honest Truth About Causal Trees: Accuracy Limits for Heterogeneous Treatment Effect Estimation Supplemental Appendix
\maketitle
\begin{abstract}
This supplemental appendix presents more general theoretical results encompassing those discussed in the main paper, and their proofs.
\end{abstract}
\textit{\small Keywords: recursive partitioning, decision trees, causal inference, heterogeneous treatment effects}
\clearpage
\setcounter{tocdepth}{3}
\tableofcontents
\clearpage
\section{Overview}\label{sec:overview}
This supplement presents proofs for the results in the main paper, and several additional theoretical results. We start with a homoskedastic constant regression model in Section~\ref{sa-sec:main}, showing that the standard CART decision tree estimator of the (constant) conditional mean suffers from slow uniform convergence rates. In Section~\ref{sa-sec:causal}, we then study the more challenging heterogeneous causal effect estimators discussed in the main paper: inverse probability weighting (IPW) estimator, the difference in mean (DIM) estimator, and the sum-of-square-minimization (SSE) estimator are considered in Sections \ref{sa-sec: ipw causal}, \ref{sa-sec: reg causal} and \ref{sa-sec: sse causal}, respectively. Section~\ref{sa-sec:proof} links the results in this supplemental appendix to those presented in the main paper.
\subsection{Notations}
\begingroup
\setlength{\parindent}{0pt}
\setlength{\parskip}{4pt}
\textbf{Sets.}
$\mathbb{R}$ is the set of real numbers and $\mathbb{N}$ the positive integers.
For $n\in\mathbb{N}$ we write $[n]=\{1,\dots ,n\}$.
\textbf{Vectors and matrices.}
Boldface lower‑case letters (e.g. $\mathbf{x}$) denote column vectors, and boldface upper‑case letters (e.g.\ $\mathbf{A}$) denote matrices.
For a vector $\mathbf{x}$, its $i$‑th component is $x_i$; for a matrix $\mathbf{A}$, its $(i,j)$‑th entry is $A_{ij}$. Denote by $\mathbf{e}_j$ the $j$-th unit vector.
\textbf{Norms.}
For $\mathbf{x}\in\mathbb{R}^d$, define $\|\mathbf{x}\| = (\sum_{i=1}^{d} x_i^{2})^{1/2}$, and $\|\mathbf{x}\|_\infty = \max_{i\le d}|x_i|$. For a matrix $A \in \mathbb{R}^{m \times n}$, the operator norm is $\lVert A \rVert = \sup_{\lVert \mathbf{x} \rVert = 1} \lVert A\mathbf{x} \rVert$, and the max norm is $\lVert A \rVert_{\max} = \max_{1 \leq i \leq m, 1 \leq j \leq n} |A_{ij}|$. For a bounded measurable function $g$,
$\|g\|_\infty = \sup_{x}|g(x)|$. For a random variable $X$ with distribution $P_X$, denote the population $L_2$ norm by $\lVert X \rVert = (\int \lVert x \rVert^2 d P_X(x))^{1/2}$; and given a random sample $\mathcal{D} = \{X_1, \cdots, X_n\}$, denote the empirical $L_2$ norm by $\lVert X \rVert_{\mathcal{D}} = (n^{-1}\sum_{i = 1}^n \lVert X_i \rVert^2)^{1/2}$.
\textbf{Asymptotics.}
For reals sequences $a_n \ll b_n$ (or $a_n = o(b_n)$) if $\limsup_{n\to\infty} \frac{|a_n|}{|b_n|} = 0$; $|a_n| \lesssim |b_n|$ (or $a_n = O(b_n)$) if there exists some constant $C$ and $N > 0$ such that $n > N$ implies $|a_n| \leq C |b_n|$. For sequences of random variables $a_n = o_{\mathbb{P}}(b_n)$ if $\operatorname{plim}_{n \rightarrow \infty}\frac{|a_n|}{|b_n|} = 0, |a_n| \lesssim_{\mathbb{P}} |b_n|$ if $\limsup_{M \rightarrow \infty} \limsup_{n \rightarrow \infty} \mathbb{P}[|\frac{a_n}{b_n}| \geq M] = 0$.
\textbf{Other.} $\mathds{1}(\cdot)$ denotes the indicator function. For two random variables $X$ and $Y$, $X \protect\mathpalette{\protect\independenT}{\perp} Y$ means $X$ and $Y$ are independent. For $x \in \mathbb{R}$, $\lfloor x \rfloor$ and $\lceil x \rceil$ denote the floor and ceiling of $x$ respectively. $\mathsf{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ denotes the Gaussian distribution with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. $\mathsf{Beta}(\alpha, \beta)$ denotes the Beta distribution with parameter $(\alpha, \beta)$. A stochastic process $\{B(t), 0 \leq t \leq 1\}$ is a Brownian bridge, if $B$ is a continuous Gaussian process with $\mathbb{E}[B(t)] = 0$, and $\mathbb{E}[B(t)B(s)] = \min\{t,s\} - ts$.
\endgroup
\subsection{Proof of Main Paper Results}\label{sa-sec:proof}
\begin{itemize}
\item \textbf{Proof of Theorem 1}: The conclusions follow from Corollary~\ref{sa-coro: uniform minimax ipw}, Corollary~\ref{sa-coro: honest output ipw}, Theorem~\ref{sa-thm: uniform minimax rates regression}, Theorem~\ref{sa-coro: honest output reg}, Corollary~\ref{sa-coro: uniform minimax rates regression fit}, and Corollary~\ref{sa-coro: honest output reg fit}.
\item \textbf{Proof of Theorem 2}: The conclusions follow from Corollary~\ref{sa-coro: L2 consistency NSS ipw}, Corollary~\ref{sa-coro: L2 consistency honest ipw}, Theorem~\ref{sa-thm: L2 consistency NSS reg}, Theorem~\ref{sa-thm: L2 consistency honest reg}, Corollary~\ref{sa-coro: L2 consistency NSS fit}, and Corollary~\ref{sa-coro: L2 consistency honest fit}.
\item \textbf{Proof of Theorem 3}: The conclusions follow from Corollary~\ref{sa-coro: main ipw}, Corollary~\ref{sa-coro: L2 consistency ipw}, Theorem~\ref{thm:main-reg}, Theorem~\ref{sa-thm: L2 consistency reg}, Corollary~\ref{sa-coro:main-reg fit}, and Corollary~\ref{sa-coro: L2 consistency fit}.
\item \textbf{Proof of Lemma 4}: See Section~\ref{sa-sec: proof of multivarite darling erdos}.
\end{itemize}
\section{Constant Regression Model}\label{sa-sec:main}
This section is self-contained, and substantially improves on the results reported in \cite{Cattaneo-Klusowski-Tian_2022_arXiv}. The results presented herein are of independent interest in regression estimation settings, and they also offer a gentle introduction to the more technically involved results discussed in Section~\ref{sa-sec:causal}.
Consider the canonical regression model where the observed data $\{(y_i,\mathbf{x}_i^T) : i = 1, 2, \dots n\}$ is a random sample satisfying
\begin{equation} \label{eq:model}
y_i = \mu(\mathbf{x}_i)+ \varepsilon_i, \qquad \mathbb{E}[\varepsilon_i \mid \mathbf{x}_i]=0, \qquad \mathbb{E}\big[\varepsilon_i^2 \mid \mathbf{x}_i\big]=\sigma^2(\mathbf{x}_i),
\end{equation}
with $\mathbf{x}_i = (x_{i1}, x_{i2}, \dots, x_{ip})^T$ a vector of $p$ covariates taking values on some support set $\mathcal{X}$.
\begin{assumption}[Location Regression Model]\label{sa-ass:DGP}
$\mathcal{D} = \{(y_i,\mathbf{x}_i^T) : 1 \leq i \leq n\}$ is a random sample such that the following conditions hold for all $i = 1,2, \cdots, n$, satisfying Equation~\eqref{eq:model} and the following:
\begin{enumerate}
\item $y_i = \mu(\mathbf{x}_i)+ \varepsilon_i$, with $\mathbb{E}[\varepsilon_i | \mathbf{x}_i]=0$ and $\mathbf{x}_i \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_i$.
\item $ \mu(\mathbf{x}) = c$ for all $\mathbf{x} \in \mathcal{X}\subseteq\mathbb{R}^p$, where $c$ is some constant.
\item $x_{i,1}, \ldots, x_{i,p}$ are independent and continuously distributed.
\item There exists $\alpha > 0$ such that $\mathbb{E}[\exp(\lambda\varepsilon_i)] < \infty$ for all $|\lambda| < 1/\alpha $ and $\sigma^2 = \mathbb{E}[\varepsilon_i^2] > 0$.
\end{enumerate}
\end{assumption}
In what follows, we denote by $P_X$ the marginal distribution of $\mathbf{x}_i$.
Now we illustrate the CART estimation strategy. Given any tree $\mathsf{T}$, the CART estimator is given as follows:
\begin{definition}[CART Estimate]\label{sa-defn: cart outcome}
Suppose $\mathsf{T}$ is the tree used, and $\mathcal{D}_{\mu}=\{(y_i, \mathbf{x}^\top_i):i=1,2,\dots,n_{\mu}\}$, with $n_{\mu}\leq n$, is the dataset used. Let $\mathsf{t}$ be the unique terminal node in $\mathsf{T}$ containing $\mathbf{x} \in \mathcal{X}$. The CART estimator is
\begin{align*}
\hat\mu(\mathbf{x}; \mathsf{T},\mathcal{D}_{\mu})
= \frac{1}{n(\mathsf{t})} \sum_{i:\mathbf{x}_i \in \mathsf{t}} y_i,
\end{align*}
where $n(\mathsf{t}) = \sum_{i=1}^{n_{\mu}} \mathds{1}(\mathbf{x}_i \in \mathsf{t})$ is the ``local'' sample sizes. In case $n(\mathsf{t}) = 0$, take $\hat\mu(\mathbf{x}; \mathsf{T},\mathcal{D}_{\mu}) = 0$.
\end{definition}
\begin{definition}[Tree Construction]\label{sa-defn: cart construction}
Given a dataset $\mathcal{D}_{\mathsf{T}} = \{(y_i,\mathbf{x}^\top_i):i=1,2,\dots,n_{\mathsf{T}}\}$, with $n_{\mathsf{T}}\leq n$, a parent node $\mathsf{t}$ in the tree (i.e., a region in $\mathcal{X}$) is divided into two child nodes, $\mathsf{t}_{\mathtt{L}}$ and $\mathsf{t}_{\mathtt{R}}$, by minimizing the sum-of-squares error (SSE),
\begin{align} \label{eq:sse}
\min_{1\leq j\leq p} \min_{\beta_{\mathtt{L}},\beta_{\mathtt{R}},\varsigma \in \mathbb{R}}
\sum_{\mathbf{x}_i \in \mathsf{t}} \big(y_{i} - \beta_{\mathtt{L}} \mathds{1}(x_{ij} \leq \varsigma) - \beta_{\mathtt{R}} \mathds{1}(x_{ij} > \varsigma) \big)^2,
\end{align}
where $(\beta_{\mathtt{L}}, \beta_{\mathtt{R}}, \varsigma, j)$ denote the two child nodes outputs, split point, and split direction, respectively. With at least one split, the final CART tree is denoted by $\mathsf{T}(\mathcal{D}_{\mathsf{T}})$.
\end{definition}
\begin{definition}[Sample Splitting]\label{sa-defn: cart sample splitting}
Recall Definition \ref{sa-defn: cart outcome} and Definition \ref{sa-defn: cart construction}, and that $\mathcal{D} = \{(y_i, \mathbf{x}^\top_i):i=1,2,\dots,n\}$ is the available random sample.
\begin{itemize}
\item \textit{No Sample Splitting} (NSS): The dataset $\mathcal{D}$ is used for both the tree construction and the treatment effect estimation, that is, $\mathcal{D}_{\mathsf{T}} = \mathcal{D}$ and $\mathcal{D}_{\mu} = \mathcal{D}$. The CART tree estimator is
\begin{align*}
\hat\mu^{\mathtt{NSS}}(\mathbf{x})
&= \hat\mu(\mathbf{x}; \mathsf{T}(\mathcal{D}),\mathcal{D}).
\end{align*}
\item \textit{Honesty} (HON): The dataset $\mathcal{D}$ is divided in two independent datasets $\mathcal{D}_{\mathsf{T}}$ and $\mathcal{D}_{\mu}$ with sample sizes $n_{\mathsf{T}}$ and $n_{\mu}$, respectively, and satisfying $n \lesssim n_{\mathsf{T}}, n_{\mu} \lesssim n$. The CART tree estimator is
\begin{align*}
\hat{\mu}^{\mathtt{HON}}(\mathbf{x})
&= \hat{\mu}(\mathbf{x}; \mathsf{T}(\mathcal{D}_{\mathsf{T}}),\mathcal{D}_{\mu}).
\end{align*}
\end{itemize}
\end{definition}
\begin{definition}[$\mathbf{X}$-Adaptive Estimation]\label{sa-defn: cart X adaptivity}
Recall Definition \ref{sa-defn: cart outcome} and Definition \ref{sa-defn: cart construction}, and that $\mathcal{D} = \{(y_i, \mathbf{x}^\top_i):i=1,2,\dots,n\}$ is the available random sample.
\begin{enumerate}
\item The dataset $\mathcal{D}$ is divided into $K+1$ datasets $(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K},\mathcal{D}_{\mu})$, with sample sizes $(n_{\mathsf{T}_1},\ldots,n_{\mathsf{T}_K},n_\mu)$, respectively, and satisfying $n_{\mathsf{T}_1} = \cdots = n_{\mathsf{T}_K} = n_{\mu}$ (possibly after dropping $n \mod K$ data points at random). For each of the datasets $\mathcal{D}_{\mathsf{T}_j} = \{(y_i, \mathbf{x}_i^{\top}):i=1,\ldots,n_{\mathsf{T}_j}\}$, $j=1,\ldots,K$, replace $\{y_i:i=1,\ldots,n_{\mathsf{T}_j}\}$ with independent copies $\{\tilde{y}_i:i=1,\ldots,n_{\mathsf{T}_j}\}$, while keeping the same $\{\mathbf{x}_i :i=1,\ldots,n_{\mathsf{T}_j} \}$.
\item The maximal decision tree of depth $K$, $\mathsf{T}_K(\mathcal{D}_{\mathsf{T}_1}, \cdots, \mathcal{D}_{\mathsf{T}_K})$, is obtained by iterating $K$ times the $l \in \{\mathtt{DIM}, \mathtt{IPW}, \mathtt{SSE}\}$ splitting procedures in Definition \ref{sa-defn: cart construction}, each time splitting \emph{all} terminal nodes until (i) the node contains a single data point $(y_i, \mathbf{x}_i^{\top})$, or (ii) the input values $\mathbf{x}_i$ and/or all $y_i$ within the node are the same.
\item The $\mathbf{X}$-adaptive estimator is
\begin{align*}
\hat\mu^{\mathtt{X}}(\mathbf{x}; K)
&= \hat\mu(\mathbf{x}; \mathsf{T}_K(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K}),\mathcal{D}_\mu).
\end{align*}
\end{enumerate}
\end{definition}
\subsection{No Sample Splitting}
We start from the no sample splitting ($\mathtt{NSS}$) case, and characterize the location of the first split.
\subsubsection*{Decision Stumps.}
For each variable $ j = 1, 2, \dots, p $, let $\pi_j$ be the permutation such that $ x_{\pi_{j}(i),j} $ is non-decreasing in the index $ i = 1, 2, \dots, n$.
Then, minimizing Equation~\eqref{eq:sse} can be equivalently recasted as maximizing the so-called \emph{impurity gain}:
\begin{equation}
\label{eq:Delta}
\begin{aligned}
& \sum_{\mathbf{x}_i \in \mathrm{t}}\big(y_i-\overline y_{\mathrm{t}}\big)^2 - \sum_{\mathbf{x}_i \in \mathrm{t}}\big(y_i - \overline y_{\mathrm{t}_L} \mathds{1}(\mathbf{x}_{i} \in \mathrm{t}_L) - \overline y_{\mathrm{t}_R} \mathds{1}(\mathbf{x}_i \in \mathrm{t}_R)\big)^2
\\
& \qquad = \frac{\Big(\frac{1}{\sqrt{n(\mathrm{t})}}\sum_{\mathbf{x}_i \in \mathrm{t}_L} (y_i-\mu) - \frac{n(\mathrm{t}_L)}{n(\mathrm{t})}\frac{1}{\sqrt{n(\mathrm{t})}}\sum_{\mathbf{x}_i \in \mathrm{t}} (y_i-\mu) \Big)^2}{(n(\mathrm{t}_L)/n(\mathrm{t}))(1-n(\mathrm{t}_L)/n(\mathrm{t}))},
\end{aligned}
\end{equation}
where $\bar{y}_{\mathrm{t}} = n(\mathrm{t})^{-1} \sum_{\mathbf{x}_i \in \mathrm{t}} y_i \mathds{1}(\mathbf{x}_i \in \mathrm{t})$.
We can show this is also equivalent to maximizing the \textit{conditional variance given the split}:
\begin{align}\label{eq:split variance}
\frac{n(t_L)n(t_R)}{n(\mathrm{t})}\big(\overline y_{\mathrm{t}_L}-\overline y_{\mathrm{t}_R}\big)^2.
\end{align}
We start by considering the case when the tree is depth one ($K=1$), i.e., a decision stump. Then optimization objectives are equivalent to choosing a splitting coordinate $\hat \jmath$, and a splitting index $\hat \imath$ such that
\begin{align*}
\mathrm{t}_L = \{\mathbf{u} \in X: \mathbf{u}_{\hat \jmath} \leq x_{\pi_{\jmath}(\imath), \jmath}\}, \qquad \mathrm{t}_R = \{\mathbf{u} \in X: \mathbf{u}_{\hat \jmath} > x_{\pi_{\jmath}(\imath), \jmath}\}.
\end{align*}
The tree output can then be written as
\begin{equation} \label{eq:stump}
\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) = \begin{cases}
\bar{y}_{\mathrm{t}_L}, \quad \mathbf{x} \in \mathrm{t}_L\\
\bar{y}_{\mathrm{t}_R}, \quad \mathbf{x} \in \mathrm{t}_R
\end{cases}\hspace{-3mm},
\end{equation}
where $x_{\hat\jmath}$ denotes the value of the $\hat\jmath$-th component of $\mathbf{x}$.
The following theorem formally (and very precisely) characterizes the regions of the support $\mathcal{X}$ where the first CART split index $ \hat\imath $, at the root node, has non-vanishing probability of realizing. As a consequence, the theorem also characterizes the effective sample size of the resulting cells (recall the data is ordered so that $ \hat \mu = x_{\hat\imath \hat\jmath} $ and hence $ \hat\imath = \#\{\mathbf{x}_i : x_{i\hat \jmath} \leq \hat \mu \} $).
\begin{theorem}[Imbalanced Splits]\label{sa-thm:master}
Suppose Assumption \ref{sa-ass:DGP} holds, and let $(\hat\imath, \hat\jmath)$ be the CART split index and split direction at the root node.
For each $ a,b \in (0, 1) $ with $ a < b $, and $\ell \in [p]$, we have
\begin{equation} \label{eq:split_range}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b}, \hat \jmath = \ell \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath \leq n-n^{a}, \hat \jmath = \ell \big) \geq \frac{b-a}{2pe},
\end{equation}
which implies
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b} \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath \leq n-n^{a} \big) \geq \frac{b-a}{2 e}.
\end{align*}
\end{theorem}
\begin{theorem}[Convergence Rates for Decision Stumps]\label{sa-thm:rates}
Suppose Assumption \ref{sa-ass:DGP} holds. Suppose the CART tree has depth $K = 1$. Then for any $a,b \in (0, 1)$ with $ a < b $, we have
\begin{equation}\label{eq:master_rate_constant}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) - \mu| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq \frac{b}{e},
\end{equation}
and suppose w.l.o.g. that $\mathbf{x}_i \thicksim \mathsf{Uniform}([0,1]^p)$, then
\begin{equation} \label{eq:master_rate_constant2}
\liminf_{n\to\infty} \inf_{\mathbf{x}\in \mathcal{X}_n} \mathbb{P}\Big(|\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) - \mu| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Big) \geq \frac{b-a}{2e},
\end{equation}
where $ \mathcal{X}_n = \{\mathbf{x} \in [0,1]^p: x_j = o(1) n^{a-1} \text{ or } 1 - x_j = o(1) n^{a-1} \text{ for some } j \in [p]\}$.
\end{theorem}
\subsubsection*{Deep Trees.}
We will show that the imbalanced split issue is inherited from the decision stumps to trees of arbitrary depth.
\begin{theorem}[Convergence Rates for Deep Trees]\label{sa-thm: uniform minimax}
Suppose Assumption \ref{sa-ass:DGP} holds. Then for any $b \in (0, 1) $, we have
$$\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) - \mu| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq b/e.$$
\end{theorem}
Therefore, decision trees grown with CART methodology cannot converge faster than any polynomial-in-$n$, when uniformity over the full support of the data $\mathcal{X}$, and over possible data generating processes, is of interest.
However, for the $L_2$-risk we still have the following positive result. This is because the small cells that leads to issues in uniform consistency will have a small measure by $\mathbb{P}_X$.
\begin{theorem}[$L_2$ Consistency -- NSS]\label{sa-thm: L2 consistency NSS}
Suppose Assumption \ref{sa-ass:DGP} holds. Then for the depth $K$ (possibly non-maximal) tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^4 \log(np)}{n},
\end{align*}
where $C$ is a positive constant that only depends on $\sigma^2$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg(\int_{\mathcal{X}} (\hat{\mu}^{\mathtt{NSS}}(\mathbf{x}) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^4 \log(np)}{n} \bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on the distribution of $\varepsilon_i$.
\end{theorem}
\subsection{Sample Splitting}
For sample splitting strategy, we also present a lower bound on uniform consistency and an upper bound on $L_2$ consistency.
\begin{theorem}\label{sa-thm: honest stump}
Suppose Assumption~\ref{sa-ass:DGP} holds. Then for any $b \in (0, 1) $, we have
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat{\mu}^{\mathtt{HON}}(\mathbf{x}) - \mu| \geq \frac{C\mathbb{E}[|y_i - \mu|]}{ n^{b/2}}\Bigg) \geq C \frac{\mathbb{E}[|y_i - \mu|^2]}{\mathbb{V}[y_i]} b,
\end{equation*}
where $C$ is some constant only depending on $\liminf_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\mu}}$ and $\limsup_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\mu}}$.
\end{theorem}
\begin{theorem}[$L_2$ Consistency -- HON]\label{sa-thm: L2 consistency honest}
Suppose Assumption \ref{sa-ass:DGP} holds. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\mu}^{\mathtt{HON}}(\mathbf{x}) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^5 }{n},
\end{align*}
provided $\rho^{-1} \leq \frac{n_{\mathsf{T}}}{n_{\mu}} \leq \rho$ for some $\rho \in (0,1)$, and $C$ is a positive constant that only depends on $\sigma^2$ and $\rho$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg(\int_{\mathcal{X}} (\hat{\mu}^{\mathtt{HON}}(\mathbf{x}) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^5}{n} \bigg) = 0,
\end{align*}
where $C^{\prime}$ is some constant only depending on $\rho$ and the distribution of $\varepsilon_i$.
\end{theorem}
Compared to Theorem~\ref{sa-thm: uniform minimax}, the lower bound on the LHS of Theorem~\ref{sa-thm: honest stump} that we characterize has one less $\sqrt{(2 + o(1)) \log \log (n)}$. Compared to Theorem~\ref{sa-thm: L2 consistency NSS}, the upper bound on the RHS of Theorem~\ref{sa-thm: L2 consistency honest} has $\log(np)$ replaced by $\log(n)$. These changes are due to the sample splitting strategy.
\subsection{X-adaptive Tree}
For X-adaptive trees, we leverage the decision stump result from Theorem~\ref{sa-thm:master} using an iterative argument to infer inconsistency of trees of depth $K_n \gtrsim \log \log (n)$.
\begin{theorem}[Pointwise Inconsistency]\label{thm:main}
Suppose Assumption \ref{sa-ass:DGP} holds. If $\liminf_{n \to \infty} \frac{K_n}{\log\log(n)} > 0$, then there exists a positive constant $C$ not depending on $n$ such that
$$
\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x} \in \mathcal{X}}|\hat{\mu}^{\mathtt{X}}(\mathbf{x}; K_n)-\mu| > C \Bigg) > 0.
$$
\end{theorem}
Since we keep the $\mathbf{x}_i$'s and refresh the $(d_i, y_i)$'s, the tree estimator has a simple form condition on $\mathbf{x}_i$'s. Hence a direct variance calculation gives us the following $L_2$-consistency result.
\begin{theorem}[L2 Consistency -- X]\label{sa-thm: L2 consistency X}
Suppose Assumption \ref{sa-ass:DGP} holds. Then
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\mu}^{\mathtt{X}}(\mathbf{x}; K) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq \frac{2^{K+1} (K+1) \sigma^2}{n + 1}.
\end{align*}
\end{theorem}
Using the same argument as Theorem~\ref{sa-thm: L2 consistency honest}, we can show
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\mu}^{\mathtt{X}}(\mathbf{x}; K) - \mu)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{K 2^K \log(n)^5}{n},
\end{align*}
where $C$ is a positive constant that only depends on $\sigma^2$. The direct variance calculation allows us to remove extra $\operatorname{poly-log}$ terms.
\section{Heterogeneous Causal Effect Estimation}\label{sa-sec:causal}
In this section, we consider the heterogeneous causal effect estimation problem from the main paper. The assumptions on the data generating process and the definitions of causal trees are the same as in the main paper. For completeness, we include them here:
\begin{assumption}[Data Generating Process]\label{sa-assump: dgp-causal}
$\mathcal{D} = \{(y_i,d_i,\mathbf{x}^\top_i): 1 \leq i \leq n\}$ is a random sample, where $y_i = d_i y_i(1) + (1 - d_i) y_i(0)$, $\mathbf{x}_i = (x_{i,1}, \ldots, x_{i,p})^{\top}$, and the following conditions hold for all $d = 0,1$ and $i = 1,2, \ldots, n$.
\begin{enumerate}
\item $(y_i(0),y_i(1),\mathbf{x}_i) \protect\mathpalette{\protect\independenT}{\perp} d_i$, and $\xi = \mathbb{P}[d_i = 1] \in (0,1)$.
\item $y_i(d) = \mu_d(\mathbf{x}_i) + \varepsilon_i(d)$, with $\mathbb{E}[\varepsilon_i(d)|\mathbf{x}_i]=0$ and $\mathbf{x}_i \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_i(d)$.
\item $\mu_d(\mathbf{x}) = c_d$ for all $\mathbf{x} \in \mathcal{X}$, where $c_d$ is some constant, and $\mathcal{X}$ is the support of $\mathbf{x}_i$.
\item $x_{i,1}, \ldots, x_{i,p}$ are independent and continuously distributed.
\item There exists $\alpha > 0$ such that $\mathbb{E}[\exp(\lambda\varepsilon_i(d))] < \infty$ for all $|\lambda| < 1/\alpha$ and $\mathbb{E}[\varepsilon_i^2(d)] > 0$.
\end{enumerate}
\end{assumption}
And the causal trees are constructed based on the following rules:
\begin{definition}[CATE Estimators]\label{sa-defn: CATE Estimators}
Suppose $\mathsf{T}$ is the tree used, and $\mathcal{D}_\tau=\{(y_i,d_i,\mathbf{x}^\top_i):i=1,2,\dots,n_{\tau}\}$, with $n_{\tau}\leq n$, is the dataset used. Let $\mathsf{t}$ be the unique terminal node in $\mathsf{T}$ containing $\mathbf{x} \in \mathcal{X}$.
\begin{itemize}
\item The \textit{Difference-in-Means} (DIM) estimator is
\begin{align*}
\hat\tau_\mathtt{DIM}(\mathbf{x}; \mathsf{T},\mathcal{D}_\tau)
= \frac{1}{n_{1}(\mathsf{t})} \sum_{i:\mathbf{x}_i \in \mathsf{t}} d_i y_i
- \frac{1}{n_{0}(\mathsf{t})} \sum_{i:\mathbf{x}_i \in \mathsf{t}} (1-d_i) y_i,
\end{align*}
where $n_{d}(\mathsf{t}) = \sum_{i=1}^{n_{\tau}} \mathds{1}(\mathbf{x}_i \in \mathsf{t}, d_i = d)$, for $d=0,1$, are the ``local'' sample sizes. In case $n_0(\mathsf{t}) = 0$ or $n_1(\mathsf{t}) = 0$, take $\hat\tau_\mathtt{DIM}(\mathbf{x}; \mathsf{T},\mathcal{D}_\tau) = 0$.
\item The \textit{Inverse Probability Weighting} (IPW) estimator is
\begin{align*}
\hat\tau_\mathtt{IPW}(\mathbf{x}; \mathsf{T},\mathcal{D}_\tau)
= \frac{1}{n(\mathsf{t})} \sum_{i:\mathbf{x}_i \in \mathsf{t}} \frac{d_i-\xi}{\xi(1-\xi)} y_i,
\end{align*}
where $n(\mathsf{t}) = n_{0}(\mathsf{t}) + n_{1}(\mathsf{t}) = \sum_{i=1}^{n_{\tau}} \mathds{1}(\mathbf{x}_i \in \mathsf{t})$ is the ``local'' sample size. In case $n(\mathsf{t}) = 0$, take $\hat\tau_\mathtt{IPW}(\mathbf{x}; \mathsf{T},\mathcal{D}_\tau) = 0$.
\end{itemize}
\end{definition}
\begin{definition}[Tree Construction]\label{sa-defn: tree construction}
Suppose $\mathcal{D}_{\mathsf{T}} = \{(y_i,d_i,\mathbf{x}^\top_i):i=1,2,\dots,n_{\mathsf{T}}\}$, with $n_{\mathsf{T}}\leq n$, is the dataset used to construct the tree $\mathsf{T}$.
\begin{itemize}
\item \textit{Variance Maximization}: A parent node $\mathsf{t}$ (i.e., a terminal node partitioning $\mathcal{X}$) in a previous tree $\mathsf{T}^{\prime}$ is divided into two child nodes, $\mathsf{t}_{\mathtt{L}}$ and $\mathsf{t}_{\mathtt{R}}$, forming the new tree $\mathsf{T}$, by maximizing
\begin{align}\label{eq: variance maximization} \frac{n(\mathsf{t}_{\mathtt{L}})n(\mathsf{t}_{\mathtt{R}})}{n(\mathsf{t})}
\Big(\hat\tau_l(\mathsf{t}_{\mathtt{L}}; \mathsf{T},\mathcal{D}_{\mathsf{T}})
- \hat\tau_l(\mathsf{t}_{\mathtt{R}}; \mathsf{T},\mathcal{D}_{\mathsf{T}})\Big)^2,
\qquad l \in \{\mathtt{DIM},\mathtt{IPW}\}.
\end{align}
With at least one split, the two final causal trees are denoted by $\mathsf{T}^{\mathtt{DIM}}(\mathcal{D}_{\mathsf{T}})$ and $\mathsf{T}^{\mathtt{IPW}}(\mathcal{D}_{\mathsf{T}})$, respectively, for $l \in \{\mathtt{DIM},\mathtt{IPW}\}$.
\item \textit{SSE Minimization}: A parent node $\mathsf{t}$ (i.e., a terminal node partitioning $\mathcal{X}$) in the previous tree $\mathsf{T}^{\prime}$ is divided into two child nodes, $\mathsf{t}_{\mathtt{L}}$ and $\mathsf{t}_{\mathtt{R}}$, forming the next tree $\mathsf{T}$, by solving
\begin{align}\label{eq: sse}
\min_{a_{\mathtt{L}}, b_{\mathtt{L}}, a_{\mathtt{R}}, b_{\mathtt{R}} \in \mathbb{R}}
\sum_{\mathbf{x}_i \in \mathsf{t}_{\mathtt{L}}} (y_i - a_{\mathtt{L}} - b_{\mathtt{L}} d_i)^2 + \sum_{\mathbf{x}_i \in \mathsf{t}_{\mathtt{R}}} (y_i - a_{\mathtt{R}} - b_{\mathtt{R}} d_i)^2,
\end{align}
where only the data $\mathcal{D}_{\mathsf{T}}$ is used. With at least one split, the final causal tree is denoted by $\mathsf{T}^{\mathtt{SSE}}(\mathcal{D}_{\mathsf{T}})$.
\end{itemize}
\end{definition}
\begin{definition}[Sample Splitting and Estimators]\label{sa-defn: data splitting}
Recall Definition \ref{sa-defn: CATE Estimators} and Definition \ref{sa-defn: tree construction}, and that $\mathcal{D} = \{(y_i, \mathbf{x}^\top_i, d_i):i=1,2,\dots,n\}$ is the available random sample.
\begin{itemize}
\item \textit{No Sample Splitting} (NSS): The dataset $\mathcal{D}$ is used for both the tree construction and the treatment effect estimation, that is, $\mathcal{D}_{\mathsf{T}} = \mathcal{D}$ and $\mathcal{D}_{\tau} = \mathcal{D}$. The causal tree estimators are
\begin{align*}
\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x})
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{DIM}}(\mathcal{D}),\mathcal{D}),\\
\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x})
&= \hat\tau_{\mathtt{IPW}}(\mathbf{x}; \mathsf{T}^{\mathtt{IPW}}(\mathcal{D}),\mathcal{D}), \quad \text{and}\\
\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x})
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{SSE}}(\mathcal{D}),\mathcal{D}),
\end{align*}
\item \textit{Honesty} (HON): The dataset $\mathcal{D}$ is divided in two independent datasets $\mathcal{D}_{\mathsf{T}}$ and $\mathcal{D}_{\tau}$ with sample sizes $n_{\mathsf{T}}$ and $n_\tau$, respectively, and satisfying $n \lesssim n_{\mathsf{T}}, n_{\tau} \lesssim n$. The causal tree estimators are
\begin{align*}
\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}(\mathbf{x})
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{DIM}}(\mathcal{D}_{\mathsf{T}}),\mathcal{D}_\tau),\\
\hat\tau^{\mathtt{HON}}_{\mathtt{IPW}}(\mathbf{x})
&= \hat\tau_{\mathtt{IPW}}(\mathbf{x}; \mathsf{T}^{\mathtt{IPW}}(\mathcal{D}_{\mathsf{T}}),\mathcal{D}_\tau), \quad \text{and}\\
\hat\tau^{\mathtt{HON}}_{\mathtt{SSE}}(\mathbf{x})
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{SSE}}(\mathcal{D}_{\mathsf{T}}),\mathcal{D}_\tau).
\end{align*}
\end{itemize}
\end{definition}
While the estimators $\hat\tau^{\mathtt{NSS}}_{l}(\mathbf{x})$ and $\hat\tau^{\mathtt{HON}}_{l}(\mathbf{x})$, $l\in\{\mathtt{DIM},\mathtt{IPW},\mathtt{SSE}\}$ depend on the depth of the tree construction used, our the notation does not make this dependence explicit because our results only require (at least) one single split.
\textbf{X-Adaptive Trees.}
\begin{definition}[$\mathbf{X}$-Adaptive Estimation]\label{sa-defn: Sample Splitting -- X-adaptivity}
Recall Definition \ref{sa-defn: CATE Estimators} and Definition \ref{sa-defn: tree construction}, and that $\mathcal{D} = \{(y_i, \mathbf{x}^\top_i, d_i):i=1,2,\dots,n\}$ is the available random sample.
\begin{enumerate}
\item The dataset $\mathcal{D}$ is divided into $K+1$ datasets $(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K},\mathcal{D}_{\tau})$, with sample sizes $(n_{\mathsf{T}_1},\ldots,n_{\mathsf{T}_K},n_\tau)$, respectively, and satisfying $n_{\mathsf{T}_1} = \cdots = n_{\mathsf{T}_K} = n_{\tau}$ (possibly after dropping $n \mod K$ data points at random). For each of the datasets $\mathcal{D}_{j} = \{(y_i, d_i, \mathbf{x}_i^{\top}):i=1,\ldots,n_{\mathsf{T}_j}\}$, $j=1,\ldots,K$, replace $\{(y_i, d_i):i=1,\ldots,n_{\mathsf{T}_j}\}$ with independent copies $\{(\tilde{y}_i, \tilde{d}_i):i=1,\ldots,n_{\mathsf{T}_j}\}$, while keeping the same $\{\mathbf{x}_i :i=1,\ldots,n_{\mathsf{T}_j} \}$.
\item The maximal decision tree of depth $K$, $\mathsf{T}^{l}_K(\mathcal{D}_{\mathsf{T}_1}, \cdots, \mathcal{D}_{\mathsf{T}_K})$, is obtained by iterating $K$ times the $l \in \{\mathtt{DIM}, \mathtt{IPW}, \mathtt{SSE}\}$ splitting procedures in Definition \ref{sa-defn: tree construction}, each time splitting \emph{all} terminal nodes until (i) the node contains a single data point $(y_i, d_i, \mathbf{x}_i^{\top})$, or (ii) the input values $\mathbf{x}_i$ and/or all $(d_i, y_i)$ within the node are the same.
\item The $\mathbf{X}$-adaptive estimators are
\begin{align*}
\hat\tau^{\mathtt{X}}_{\mathtt{DIM}}(\mathbf{x}; K)
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{DIM}}_{K}(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K}),\mathcal{D}_\tau),\\
\hat\tau^{\mathtt{X}}_{\mathtt{IPW}}(\mathbf{x}; K)
&= \hat\tau_{\mathtt{IPW}}(\mathbf{x}; \mathsf{T}^{\mathtt{IPW}}_{K}(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K}),\mathcal{D}_\tau), \quad \text{and}\\
\hat\tau^{\mathtt{X}}_{\mathtt{SSE}}(\mathbf{x}; K)
&= \hat\tau_{\mathtt{DIM}}(\mathbf{x}; \mathsf{T}^{\mathtt{SSE}}_{K}(\mathcal{D}_{\mathsf{T}_1},\ldots,\mathcal{D}_{\mathsf{T}_K}),\mathcal{D}_\tau).
\end{align*}
\end{enumerate}
\end{definition}
\subsection{IPW Estimator}\label{sa-sec: ipw causal}
The transformed outcomes $y_i \frac{d_i - \xi}{\xi (1 - \xi)}, 1 \leq i \leq n,$ are i.i.d, with
\begin{align*}
\mathbb{E} \bigg[y_i \frac{d_i - \xi}{\xi (1 - \xi)}\bigg|\mathbf{x}_i\bigg] = \mathbb{E}[y_i(1) - y_i(0)|\mathbf{x}_i] = c_1 - c_0,
\end{align*}
and
\begin{align*}
\tilde{\varepsilon}_i = y_i \frac{d_i - \xi}{\xi(1 - \xi)} - (c_1 - c_0)
& = (c_1 + \varepsilon_i(1)) \frac{d_i}{\xi} - (c_0 + \varepsilon_i(0)) \frac{1 - d_i}{1 - \xi} - (c_1 - c_0) \protect\mathpalette{\protect\independenT}{\perp} \mathbf{x}_i.
\end{align*}
Assumption~\ref{sa-assump: dgp-causal} implies $\mathbb{E}[\exp(\lambda \tilde{\varepsilon}_i)] < \infty$ for all $|\lambda| \leq 1 / \beta$ with $\beta$ only depending on $\xi$ and $\alpha$, and $\mathbb{E}[\tilde{\varepsilon}_i^2] >0$. Hence the following results are immediate corollaries from the results in Section~\ref{sa-sec:main}.
\subsubsection{No Sample Splitting}
\begin{coro}[Imbalanced Split]\label{coro: inconsistency ipw}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Then for each $a,b \in (0,1)$ with $a < b$, for every $\ell \in [p]$,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b}, \hat \jmath = \ell \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath \leq n-n^{a}, \hat \jmath = \ell \big) \geq \frac{b-a}{2pe}.
\end{equation*}
\end{coro}
\begin{coro}[Stump]\label{coro: rates ipw}
Suppose Assumption \ref{sa-assump: dgp-causal} holds, and the tree has depth $K = 1$. Then for any $ a,b \in (0, 1) $ with $ a < b $, we have
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq \frac{b}{e},
\end{equation*}
where $\sigma^2 = \mathbb{V} \Big[\frac{d_i y_i(1)}{\xi} + \frac{(1 - d_i) y_i(0)}{1 - \xi}\Big]$. Moreover, if $\mathbf{x}_i$ has a density that is continuous and positive on $[0,1]^p$, then
\begin{equation*}
\liminf_{n\to\infty} \inf_{\mathbf{x}\in \mathcal{X}_n} \mathbb{P}\Big(|\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Big) \geq \frac{b-a}{2e},
\end{equation*}
where $ \mathcal{X}_n = \{\mathbf{x} \in [0,1]^p: x_j = o(1) n^{a-1} \text{ or } 1 - x_j = o(1) n^{a-1} \text{ for some } j \in [p]\}$.
\end{coro}
\begin{coro}[Rates]\label{sa-coro: uniform minimax ipw}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Then for any $b \in (0, 1) $ and arbitrary depth tree, we have
$$\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq \frac{b}{e}.$$
\end{coro}
\begin{coro}[$L_2$ Consistency -- NSS]\label{sa-coro: L2 consistency NSS ipw}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^4 \log(np)}{n},
\end{align*}
where $C$ is a positive constant that only depends on the distribution of $\tilde{\varepsilon}_i = y_i \frac{d_i - \xi}{\xi(1 - \xi)} - \tau$. Moreover,
\begin{align*}
\limsup_{n \to \infty}\mathbb{P} \bigg(\int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{IPW}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^4 \log(np)}{n}\bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on the distribution of $\tilde{\varepsilon}_i$.
\end{coro}
\subsubsection{Sample Splitting}
\begin{coro}[Honest Causal Output]\label{sa-coro: honest output ipw}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Then for any $b \in (0, 1) $, we have
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{HON}}_{\mathtt{IPW}}(\mathbf{x}) - \tau| \geq \frac{C\mathbb{E}[|\tilde{\varepsilon}_i|]}{8 n^{b/2}}\Bigg) \geq C \frac{\mathbb{E}[|\tilde{\varepsilon}_i|^2]}{\mathbb{V}[\tilde{\varepsilon}_i]} b,
\end{equation*}
where $C$ is some constant only depending on the distribution of $\tilde{\varepsilon}_i = y_i \frac{d_i - \xi}{\xi(1 - \xi)} - \tau$, $\liminf_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$ and $\limsup_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$.
\end{coro}
\begin{coro}[$L_2$ Consistency -- HON]\label{sa-coro: L2 consistency honest ipw}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{IPW}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^5}{n},
\end{align*}
provided $\rho^{-1} \leq \frac{n_{\mathsf{T}}}{n_{\tau}} \leq \rho$ for some $\rho \in (0,1)$, and $C$ is some constant only depending on the distribution of $\tilde{\varepsilon}_i = y_i \frac{d_i - \xi}{\xi(1 - \xi)} - \tau$ and $\rho$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg( \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{IPW}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^5}{n} \bigg) = 0,
\end{align*}
where $C^{\prime}$ is some constant only depending on the distribution of $\tilde{\varepsilon}_i$ and $\rho$.
\end{coro}
\subsubsection{X-adaptive Tree}
\begin{coro}[Honest CART+]\label{sa-coro: main ipw}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Suppose $\liminf_{n \to \infty} \frac{K_n}{\log \log (n)} > 0$. Then, there exists a positive constant $C$ not depending on $n$ such that
\begin{align*}
\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x} \in \mathcal{X}}|\hat\tau^{\mathtt{X}}_{\mathtt{IPW}}(\mathbf{x}; K_n)-\tau| > C \Bigg) > 0.
\end{align*}
\end{coro}
\begin{coro}[L2 Consistency -- X]\label{sa-coro: L2 consistency ipw}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{X}}_{\mathtt{IPW}}(\mathbf{x}; K) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^{K} K \sigma^2}{n},
\end{align*}
where $C$ is some constant only depending on the distribution of $\tilde{\varepsilon}_i = y_i \frac{d_i - \xi}{\xi(1 - \xi)} - \tau$.
\end{coro}
\subsection{DIM Estimator}\label{sa-sec: reg causal}
The $\mathtt{DIM}$ estimator can not be directly written as a regression tree with transformed outcome. However, we show that it can be approximated by an $\mathtt{IPW}$-tree. More specifically, we view the split criterion with different splitting index and coordinate as an empirical process, and show that the split criterion for $\mathtt{DIM}$ and $\mathtt{IPW}$ approximate each other.
\subsubsection{No Sample Splitting}
\subsubsection*{Approximation Results on Decision Stumps.}
Denote by $\pi_{\ell}$ permutation of index $[n]$ such that $x_{\pi_{\ell}(1),\ell} \leq x_{\pi_{\ell}(2),\ell} \leq \cdots \leq x_{\pi_{\ell}(n),\ell}$, $1 \leq \ell \leq p$. Consider the split criterion for the regression and ipw trees when splitting at the root note when $\#\{\mathbf{x}_{\pi_{\ell}(i)} \in t_L\} = k$: For $1 \leq \ell \leq p$, $1 \leq k \leq n$, consider
\begin{align*}
\mathscr{I}^{\mathtt{DIM}}(k,\ell) = \frac{k(n-k)}{n}\Big(\hat\tau^{\mathtt{DIM}}_{L}(k,\ell)-\hat\tau^{\mathtt{DIM}}_{R}(k,\ell)\Big)^2,\\
\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell) = \frac{k(n-k)}{n}\Big(\bar\tau^{\mathtt{IPW}}_{L}(k,\ell)-\bar\tau^{\mathtt{IPW}}_{R}(k,\ell)\Big)^2,
\end{align*}
where
\begin{align*}
\hat\tau^{\mathtt{DIM}}_{L}(k,\ell) & = \frac{\sum_{i = 1}^k d_{\pi_{\ell}(i)} y_{\pi_{\ell}(i)}}{\sum_{i = 1}^k d_{\pi_{\ell}(i)}} - \frac{\sum_{i = 1}^k (1 - d_{\pi_{\ell}(i)}) y_{\pi_{\ell}(i)}}{\sum_{i = 1}^k (1 - d_{\pi_{\ell}(i)})}, \\
\hat\tau^{\mathtt{DIM}}_{R}(k,\ell) & = \frac{\sum_{i = k+1}^n d_{\pi_{\ell}(i)} y_{\pi_{\ell}(i)}}{\sum_{i = k+1}^n d_{\pi_{\ell}(i)}} - \frac{\sum_{i = k+1}^n (1 - d_{\pi_{\ell}(i)}) y_{\pi_{\ell}(i)}}{\sum_{i = k+1}^n (1 - d_{\pi_{\ell}(i)})}, \\
\bar\tau^{\mathtt{IPW}}_{L}(k,\ell) & = \frac{1}{k}\sum_{i = 1}^k \frac{d_{\pi_{\ell}(i)}}{\xi} \varepsilon_{\pi_{\ell}(i)}(1) - \frac{1}{k}\sum_{i = 1}^k \frac{1 - d_{\pi_{\ell}(i)}}{1 - \xi} \varepsilon_{\pi_{\ell}(i)}(0), \\
\bar\tau^{\mathtt{IPW}}_{R}(k,\ell) & = \frac{1}{n-k}\sum_{i = k+1}^n \frac{d_{\pi_{\ell}(i)}}{\xi} \varepsilon_{\pi_{\ell}(i)}(1)- \frac{1}{n-k}\sum_{i = k+1}^n \frac{1 - d_{\pi_{\ell}(i)}}{1 - \xi} \varepsilon_{\pi_{\ell}(i)}(0).
\end{align*}
Notice that if we replace $\varepsilon_{\pi_{\ell}(i)}$ by $y_{\pi_{\ell}(i)}$, we would get $\hat \tau_L^\mathtt{IPW}$ (or $\hat \tau_R^\mathtt{IPW}$) instead of $\bar \tau_L^\mathtt{IPW}$ (or $\bar \tau_R^\mathtt{IPW}$). But putting $\varepsilon_{\pi_\ell(i)}$ here allows us to approximate the $\mathscr{I}^\mathtt{DIM}(\cdot, \ell)$ processes.
The optimization objective based on Definition~\ref{sa-defn: tree construction} for the regression based estimator with variance maximization is equivalent to choosing a splitting coordinate $\hat \jmath_{\text{DIM}}$, and a splitting index $\hat \imath_{\text{DIM}}$ such that
\begin{align*}
\mathrm{t}_L = \{\mathbf{u} \in \mathcal{X}: \mathbf{u}_{\hat \jmath_\text{DIM}} \leq x_{\pi_{\hat \jmath_\text{DIM}}(\hat \imath_\text{DIM}), \hat \jmath_\text{DIM}}\}, \qquad \mathrm{t}_R = \{\mathbf{u} \in \mathcal{X}: \mathbf{u}_{\hat \jmath_\text{DIM}} > x_{\pi_{\hat \jmath_\text{DIM}}(\hat \imath_\text{DIM}), \hat \jmath_\text{DIM}}\},
\end{align*}
that maximizes
\begin{align*}
\frac{n(t_L)n(t_R)}{n(\mathrm{t})}\Big(\hat\tau_{\text{DIM}}(t_L)-\hat\tau_{\text{DIM}}(t_R)\Big)^2,
\end{align*}
that is,
\begin{align*}
(\hat{\imath}_\text{DIM}, \hat{\jmath}_\text{DIM}) = \operatorname*{arg\,max}_{k,\ell}\mathscr{I}^{\mathtt{DIM}}(k,\ell).
\end{align*}
A technical aspect is to control for fluctuations of objects of the form $\frac{\sum_{i = 1}^k d_{\pi_{\ell}(i)} y_{\pi_{\ell}(i)}}{\sum_{i = 1}^k d_{\pi_{\ell}(i)}}$, for which we will use a truncation argument that requires $\sum_{i = 1}^k d_{\pi_{\ell}(i)} \geq r_n$ with $r_n \to \infty$. This gives the following lemma:
\begin{lemma}[Approximation Error]\label{lem: approximation -- balanced region}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Let $(r_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $r_n \rightarrow \infty$. Then
\begin{align*}
\max_{1 \leq \ell \leq p}\max_{r_n \leq k < n - r_n} \Big|\mathscr{I}^{\mathtt{DIM}}(k,\ell) - \bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)\Big| = O_{\mathbb{P}} \bigg(\frac{\log \log (n)}{\sqrt{r_n}}\bigg).
\end{align*}
\end{lemma}
We also control for the truncation error:
\begin{lemma}[Truncation Error]\label{lem: approximation -- imbalanced region}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Let $\rho_n$ be a sequence taking values in $(0,1)$ such that $\limsup_{n \rightarrow \infty} \rho_n \log \log (n) = 0$, and take $s_n = \exp((\log n)^{\rho_n})$. Then
\begin{align*}
\max_{1 \leq \ell \leq p}\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \Big|\mathscr{I}^{\mathtt{DIM}}(k,\ell) - \bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)\Big| = O_\mathbb{P}\bigg(\rho_n \log \log (n) + \frac{s_n}{n - s_n} \log \log(n)\bigg).
\end{align*}
\end{lemma}
\subsubsection*{Rates for Decision Stumps.}
The previous two lemmas imply that we can study $\operatorname*{arg\,max}$ of $\mathscr{I}^\mathtt{DIM}$ in terms of $\operatorname*{arg\,max}$ of $\bar \mathscr{I}^\mathtt{IPW}$. The latter is the split criterion based on CART with \emph{transformed outcome} $\frac{d_i}{\xi} \varepsilon_i(1) - \frac{1 - d_i}{1 - \xi} \varepsilon_i(0)$, and results from Section~\ref{sa-sec:main} can be applied.
\begin{theorem}[Imbalanced Split]\label{sa-thm: imbalance reg}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Then for each $a,b \in (0,1)$ with $a < b$, for every $\ell \in [p]$,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath_\mathtt{DIM} \leq n^{b}, \hat \jmath_\mathtt{DIM} = \ell \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath_\mathtt{DIM} \leq n-n^{a}, \hat \jmath_\mathtt{DIM} = \ell \big) \geq \frac{b-a}{2pe}.
\end{equation*}
\end{theorem}
The issue of imbalanced cells gives rise to the slow uniform convergence rate.
\begin{theorem}[Rates for Stump]\label{sa-thm:rates_reg}
Suppose Assumption \ref{sa-assump: dgp-causal} holds, and the tree has depth $K = 1$. Then for any $ a,b \in (0, 1) $ with $ a < b $,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq \frac{b}{e},
\end{equation*}
where $\sigma^2 = \mathbb{V}[\tilde{\varepsilon_i}]$, with $\tilde{\varepsilon}_i = \frac{d_i}{\xi} \varepsilon_i(1) - \frac{1 - d_i}{1 - \xi} \varepsilon_i(0)$. Suppose w.l.o.g. that $\mathbf{x}_i \thicksim \mathsf{Uniform}([0,1]^p)$, then
\begin{equation*}
\liminf_{n\to\infty} \inf_{\mathbf{x}\in \mathcal{X}_n} \mathbb{P}\Big(|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Big) \geq \frac{b-a}{2e},
\end{equation*}
where $\mathcal{X}_n = \{\mathbf{x} \in [0,1]^p: x_j = o(1) n^{a-1} \text{ or } 1 - x_j = o(1) n^{a-1} \text{ for some } j \in [p]\}$.
\end{theorem}
\subsubsection*{Deeper Trees.}
We generalize the above results on decision stumps to trees of arbitrary depths.
\begin{theorem}[Deeper Trees]\label{sa-thm: uniform minimax rates regression}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then for any $b \in (0, 1) $,
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq b/e.
\end{align*}
\end{theorem}
In comparison to the uniform convergence rate, for $L_2$ convergence rate we can give an upper bound as follows.
\begin{theorem}[$L_2$ Consistency -- NSS]\label{sa-thm: L2 consistency NSS reg}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^4 \log(np)}{n},
\end{align*}
where $C$ is a positive constant that only depends on the distribution of $(d_i, \varepsilon_i(0), \varepsilon_i(1))$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg( \int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^4 \log(np)}{n} \bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on the distribution of $(d_i, \varepsilon_i(0), \varepsilon_i(1))$.
\end{theorem}
\subsubsection{Sample Splitting}
With the sample splitting strategy, we also give a lower bound on uniform convergence rate and an upper bound on $L_2$ convergence rate. The difference in rates from the rates in the previous section is due to the different sample splitting strategies.
\begin{theorem}[Honest Causal Output]\label{sa-coro: honest output reg}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Then for any $b \in (0, 1) $,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq C n^{-b/2} \Bigg) \geq C \xi(1 - \xi) b.
\end{equation*}
where $C$ is some positive constant only depending on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$, $\liminf_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$ and $\limsup_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$.
\end{theorem}
\begin{theorem}[$L_2$ Consistency -- HON]\label{sa-thm: L2 consistency honest reg}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^5}{n},
\end{align*}
provided $\rho^{-1} \leq \frac{n_{\mathsf{T}}}{n_{\tau}} \leq \rho$ for some $\rho \in (0,1)$, and $C$ is a positive constant that only depends on $\rho$ and the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg( \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^5}{n} \bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on $\rho$ and the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$.
\end{theorem}
\subsubsection{X-adaptive Tree}
We leverage Theorem~\ref{sa-thm: imbalance reg} with an iterative argument to get
\begin{theorem}[CART+]\label{thm:main-reg}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Suppose $\liminf_{n \to \infty} \frac{K_n}{\log \log(K_n)} > 0$. Then
\begin{align*}
\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x} \in \mathcal{X}}|\hat{\tau}^{\mathcal{X}}_{\mathtt{DIM}}(\mathbf{x}; K_n)-\tau| > C \Bigg) > 0,
\end{align*}
where $C$ is some positive constant not depending on $n$.
\end{theorem}
A direct variance calculation gives
\begin{theorem}[L2 Consistency]\label{sa-thm: L2 consistency reg}
Suppose Assumption \ref{sa-assump: dgp-causal} holds. Then
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\tau}^{\mathcal{X}}_{\mathtt{DIM}}(\mathbf{x}; K) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{K \; 2^{K}}{n},
\end{align*}
where $C$ is some positive constant that only depends on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$.
\end{theorem}
Using the same argument as Theorem~\ref{sa-thm: L2 consistency honest reg}, we can show
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat{\tau}^{\mathcal{X}}_{\mathtt{DIM}}(\mathbf{x}; K) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{K 2^K \log(n)^5}{n},
\end{align*}
where $C$ is a positive constant that only depends on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$. The direct variance calculation allows us to remove extra $\operatorname{poly-log}$ terms.
\subsection{SSE Estimator}\label{sa-sec: sse causal}
While the CATE estimators given the tree of the $\mathtt{SSE}$ strategy coincides with the $\mathtt{DIM}$ strategy, the tree construction methods differ. Similar to $\mathtt{DIM}$, for $\mathtt{SSE}$ we also characterize the distribution of split index via a Gaussian approximation. Here we show the split criterion with $\mathtt{SSE}$ strategy can be approximated by the split criterion from two transformed outcome regressions, one for treatment and one for control. A careful high dimensional Gaussian approximation with respect to the geometry of simple convex sets then enables us to characterize the limiting distribution of splitting indices.
\subsubsection{No Sample Splitting}
\subsubsection*{Decision Stump.}
For each variable $ j = 1, 2, \dots, p $, the data $ \{ x_{ij}: \mathbf{x}_i \in \mathrm{t} \} $ is relabeled so that $ x_{ij} $ is increasing in the index $ i = 1, 2, \dots, n(\mathrm{t}) $, where $ n(\mathrm{t}) = \#\{ \mathbf{x}_i \in \mathrm{t} \} $. The fit-based objective is to minimize
\begin{align}\label{eq:fit-based}
\min_{a_L, b_L, a_R, b_R \in \mathbb{R}} \sum_{\mathbf{x}_i \in t_L} (y_i - a_{t_L} - b_{t_L} d_i)^2 + \sum_{\mathbf{x}_i \in t_R} (y_i - a_{t_R} - b_{t_R} d_i)^2
\end{align}
with respect to the index $ i $ and variable $ j $. Again, the maximizers are denoted by $ (\hat\imath_\text{SSE}, \hat\jmath_\text{SSE}) $, and the optimal split point $ \hat \tau $ that maximizes \eqref{eq:fit-based} can be expressed as $ x_{\hat\imath_\text{SSE}, \hat\jmath_\text{SSE}}$.
To break down the criterion \eqref{eq:fit-based}, denote
\begin{align*}
& \hat \mu_{L,0}(k,\ell) = \frac{\sum_{i = 1}^k (1 - d_{\pi_{\ell}(i)}) y_{\pi_{\ell}(i)}}{\sum_{i = 1}^k (1 - d_{\pi_{\ell}(i)})}, && \hat \mu_{L,1}(k,\ell) = \frac{\sum_{i = 1}^k d_{\pi_{\ell}(i)} y_{\pi_{\ell}(i)}}{\sum_{i = 1}^k d_{\pi_{\ell}(i)}}, \\
& \hat \mu_{R,0}(k,\ell) = \frac{\sum_{i = k+1}^n (1 - d_{\pi_{\ell}(i)}) y_{\pi_{\ell}(i)}}{\sum_{i = k+1}^n (1 - d_{\pi_{\ell}(i)})}, && \hat \mu_{R,1}(k,\ell) = \frac{\sum_{i = k+1}^n d_{\pi_{\ell}(i)} y_{\pi_{\ell}(i)}}{\sum_{i = k+1}^n d_{\pi_{\ell}(i)}}.
\end{align*}
Also to denote the counts compactly, $n_0 = \sum_{i = 1}^n (1 - d_i)$, $n_{L,0}(k) = \sum_{i = 1}^k (1 - d_{\pi_{\ell}(i)})$, $n_{R,0}(k) = \sum_{i = k + 1}^n (1 - d_{\pi_{\ell}(i)})$, and $n_1 = \sum_{i = 1}^n d_i$, $n_{L,1}(k) = \sum_{i = 1}^k d_{\pi_{\ell}(i)}$, $n_{R,1}(k) = \sum_{i = k + 1}^n d_{\pi_{\ell}(i)}$. Then we can show that maximizing Equation~\eqref{eq:fit-based} is equivalent to maximizing
\begin{align*}
\mathscr{I}^\text{SSE}(k, \ell) = \frac{n_{L,0} n_{R,0}}{n_0} (\hat \mu_{L,0}(k,\ell) - \hat \mu_{R,0}(k,\ell))^2 + \frac{n_{L,1} n_{R,1}}{n_1} (\hat \mu_{L,1}(k,\ell) - \hat \mu_{R,1}(k,\ell))^2.
\end{align*}
We want to show the above empirical process can be approximated by
\begin{align*}
\mathscr{I}^\text{prox}(k, \ell) = & (1 - \xi) \frac{k (n - k)}{n} (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2 + \xi \frac{k (n - k)}{n} (\bar \mu_{L,1}(k,\ell) - \bar \mu_{R,1}(k,\ell))^2,
\end{align*}
with
\begin{align*}
& \bar \mu_{L,0}(k,\ell) = \frac{1}{k}\sum_{i \leq k} \frac{1 - d_{\pi_\ell(i)}}{1 - \xi} Y_{\pi_\ell(i)}, && \bar \mu_{L,1}(k,\ell) = \frac{1}{k}\sum_{i \leq k} \frac{d_{\pi_\ell(i)}}{\xi} Y_{\pi_\ell(i)}, \\
& \bar \mu_{R,0}(k,\ell) = \frac{1}{n - k}\sum_{i > k} \frac{1 - d_{\pi_\ell(i)}}{1 - \xi} Y_{\pi_\ell(i)}, && \bar \mu_{R,1}(k,\ell) = \frac{1}{n - k}\sum_{i > k} \frac{d_{\pi_\ell(i)}}{\xi} Y_{\pi_\ell(i)}.
\end{align*}
The latter can be approximated by the summation of two independent time-transformed O-U process (which is again a time-transformed O-U process), for fixed coordinate $\ell \in [p]$. More precisely, we present the approximation lemmas:
\begin{lemma}[Approximation Error]\label{lem: approximation fit based -- balanced region}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Let $(r_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $r_n \rightarrow \infty$. Then
\begin{align*}
\max_{1 \leq \ell \leq p}\max_{r_n \leq k < n - r_n} \Big|\mathscr{I}^{\text{SSE}}(k,\ell) - \mathscr{I}^{\text{prox}}(k,\ell)\Big| = O_{\mathbb{P}} \bigg(\frac{\log \log (n)^{3/2}}{\sqrt{r_n}}\bigg).
\end{align*}
\end{lemma}
\begin{lemma}[Truncation Error]\label{lem: approximation fit based -- imbalanced region}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds. Let $\rho_n$ be a sequence taking values in $(0,1)$ such that $\limsup_{n \rightarrow \infty} \rho_n \log \log (n) = \infty$, and take $s_n = \exp((\log n)^{\rho_n})$. Then
\begin{align*}
\max_{1 \leq \ell \leq p}\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \Big|\mathscr{I}^{\text{SSE}}(k,\ell) - \mathscr{I}^{\text{prox}}(k,\ell)\Big| = O_\mathbb{P}\bigg(\rho_n \log \log (n) + \frac{s_n}{n - s_n} \log \log(n)\bigg).
\end{align*}
\end{lemma}
\begin{theorem}\label{thm: inconsistency fit}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then for each $a,b \in (0,1)$ with $a < b$, for every $\ell \in [p]$,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath_\text{SSE} \leq n^{b}, \hat \jmath_\text{SSE} = \ell \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath_\text{SSE} \leq n-n^{a}, \hat \jmath_\text{SSE} = \ell \big) \geq \frac{b-a}{2pe}.
\end{equation*}
\end{theorem}
\begin{remark}
We add the condition that $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$ so that a two-dimensional Darling-Erdos theorem \cite[Lemma 2.1]{horvath1993maximum} can be applied. We conjecture that without $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$, the conclusion still holds with a Darling-Erdos theorem for i.n.i.d O-U process, but this is out of the scope of this paper.
\end{remark}
Notice that although the splitting criteria is different from the regression tree, once cells are given the estimator given by the fit-based tree is exactly the same as the regression tree (see Section~\ref{sa-sec: reg causal}). Hence the following results can be proved based on Theorem~\ref{thm: inconsistency fit} and the same logic as Theorem~\ref{sa-thm:rates_reg} to Theorem~\ref{thm:main-reg}.
\begin{coro}[Rates for Stump]\label{sa-coro:rates_reg_fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. For any $ a,b \in (0, 1) $ with $ a < b $, we have
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq \frac{b}{e},
\end{equation*}
and suppose w.l.o.g. that $\mathbf{x}_i \thicksim \mathsf{Uniform}([0,1]^p)$, then
\begin{equation*}
\liminf_{n\to\infty} \inf_{\mathbf{x}\in \mathcal{X}_n} \mathbb{P}\Big(|\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Big) \geq \frac{b-a}{2e},
\end{equation*}
where $\mathcal{X}_n = \{\mathbf{x} \in [0,1]^p: x_j = o(1) n^{a-1} \text{ or } 1 - x_j = o(1) n^{a-1} \text{ for some } j \in [p]\}$, and $\sigma^2 = \mathbb{V}[\frac{d_i y_i(1)}{\xi} + \frac{(1 - d_i) y_i(0)}{1 - \xi}]$.
\end{coro}
\subsubsection*{Deeper Trees.}
\begin{coro}[Deeper Trees]\label{sa-coro: uniform minimax rates regression fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then for any $b \in (0, 1) $, for any sequence $K_n$ taking values in $\mathbb{N}$,
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Bigg) \geq b/e.
\end{align*}
\end{coro}
\begin{coro}[$L_2$ Consistency -- NSS]\label{sa-coro: L2 consistency NSS fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^4 \log(np)}{n},
\end{align*}
where $C$ is a positive constant that only depends on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg( \int_{\mathcal{X}} (\hat\tau^{\mathtt{NSS}}_{\mathtt{SSE}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^4 \log(np)}{n}\bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$.
\end{coro}
\subsubsection{Sample Splitting}
\begin{coro}[Honest Causal Output]\label{sa-coro: honest output reg fit}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then for any $b \in (0, 1) $, for any sequence $K_n$ taking values in $\mathbb{N}$,
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\Bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{HON}}_{\mathtt{SSE}}(\mathbf{x}) - \tau| \geq C n^{-b/2} \Bigg) \geq C \xi(1 - \xi) b.
\end{equation*}
where $C$ is some constant only depending on the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$, and $\liminf_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$ and $\limsup_{n \to \infty} \frac{n_{\mathsf{T}}}{n_{\tau}}$.
\end{coro}
\begin{coro}[$L_2$ Consistency -- HON]\label{sa-coro: L2 consistency honest fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then for the depth $K$ (possibly non-maximal) causal tree,
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{SSE}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{2^K \log(n)^5}{n},
\end{align*}
provided $\rho^{-1} \leq \frac{n_{\mathsf{T}}}{n_{\tau}} \leq \rho$ for some $\rho \in (0,1)$, and $C$ is a positive constant that only depends on $\rho$ and the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$. Moreover,
\begin{align*}
\limsup_{n \to \infty} \mathbb{P} \bigg( \int_{\mathcal{X}} (\hat\tau^{\mathtt{HON}}_{\mathtt{SSE}}(\mathbf{x}) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x}) \geq C^{\prime} \frac{2^K \log(n)^5}{n}\bigg) = 0,
\end{align*}
where $C^{\prime}$ is a positive constant that only depends on $\rho$ and the distribution of $(\varepsilon_i(0), \varepsilon_i(1), d_i)$.
\end{coro}
\subsubsection{X-adaptive Tree}
\begin{coro} \label{sa-coro:main-reg fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Suppose $\liminf_{n \to \infty} \frac{K_n}{\log \log(n)} > 0$. Then
\begin{align*}
\liminf_{n\to\infty}\mathbb{P}\Bigg(\sup_{\mathbf{x} \in \mathcal{X}}|\hat\tau^{\mathtt{X}}_{\mathtt{SSE}}(\mathbf{x}; K_n)-\tau| > C \Bigg) > 0.
\end{align*}
\end{coro}
\begin{coro}[L2 Consistency]\label{sa-coro: L2 consistency fit}
Suppose Assumption \ref{sa-assump: dgp-causal} holds with $\mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$. Then
\begin{align*}
\mathbb{E} \bigg[ \int_{\mathcal{X}} (\hat\tau^{\mathtt{X}}_{\mathtt{SSE}}(\mathbf{x}; K) - \tau)^2 dF_{\mathbf{X}}(\mathbf{x})\bigg] \leq C \frac{K \; 2^{K}}{n},
\end{align*}
where $C$ is some constant not depending on $n$.
\end{coro}
\subsection{Additional Results}
\subsubsection{Squared T-statistic Estimators}
The fourth method proposed by \cite{Athey-Imbens_2016_PNAS} is the squared T-statistic trees, where at the root node the index and coordinate to split $(\hat \imath, \hat \jmath)$ are chosen so that the \emph{squared T-statistics metric} is maximized, that is,
\begin{align*}
\hat \imath, \hat \jmath = \operatorname*{arg\,max}_{k \in [n], \ell \in [p]} n \frac{(\hat \tau_L(k,\ell) - \hat \tau_R(k, \ell))^2}{S(k, \ell)^2/k + S(k,\ell)^2/(n - k)},
\end{align*}
where $\hat{\tau}_L(k,\ell)$ and $\hat{\tau}_R(k,\ell)$ are the causal tree estimators for the left and right nodes respectively based on split coordinate $\ell$ and index $k$, and $S(k,\ell)^2$ is the conditional sample variance given the split, that is,
\begin{align*}
S(k,\ell)^2 & = \frac{1}{n - 2} \sum_{i \leq k} (\tau_i - \hat \tau_L(k,\ell))^2 + \frac{1}{n - 2} \sum_{i > k} (\tau_i - \hat \tau_R(k, \ell))^2 \\
& = \frac{1}{n - 2} \bigg[\sum_{i = 1}^n (\tau_i - n^{-1}\sum_{j = 1}^n \tau_i)^2 - \frac{k (n - k)}{n} (\hat \tau_L(k,\ell) - \hat \tau_R(k, \ell))^2\bigg].
\end{align*}
Putting together, we see the \emph{squared T-statistics metric} is a monotone transformation of the split criterion of previously studied estimators,
\begin{align*}
& n \frac{(\hat \tau_L(k,\ell) - \hat \tau_R(k, \ell))^2}{S(k,\ell)^2/k + S(k,\ell)^2/(n - k)} \\
& = n \frac{k (n - k)}{n} (\hat \tau_L(k,\ell) - \hat \tau_R(k, \ell))^2 \bigg(\frac{1}{n - 2}\sum_{i = 1}^n (\tau_i - n^{-1}\sum_{j = 1}^n \tau_i)^2 - \frac{1}{n-2} \frac{k (n - k)}{n} (\hat \tau_L(k,\ell) - \hat \tau_R(k, \ell))^2\bigg)^{-1}.
\end{align*}
Hence the split is always the same as the split by the split criterion studied in Section~\ref{sa-sec: ipw causal} and Section~\ref{sa-sec: reg causal}.
\subsubsection{Unbiasedness under Symmetric Error}\label{sa-sec: unbiased}
\begin{lemma}[Unbiasedness]\label{sa-lem: unbiased}
Suppose Assumption~\ref{sa-assump: dgp-causal} holds, and $\varepsilon_i(0)$, $\varepsilon_i(1)$ are symmetrically distributed around zero. Then
\begin{align*}
\mathbb{E}[\hat\tau^{q}_{l}(\mathbf{x}; K)] = \tau,
\qquad
l \in \{\mathtt{DIM},\mathtt{IPW},\mathtt{SSE}\},
\quad
q \in \{\mathtt{NSS},\mathtt{X}\}, \quad K \geq 1,
\end{align*}
and suppose $\mathsf{t}$ is the node containing $\mathbf{x}$, then
\begin{align*}
& \mathbb{E}[\hat\tau_l^{\mathtt{HON}}(\mathbf{x};K)] = \tau - \tau \mathbb{P}(n(\mathsf{t}) = 0), \qquad l \in \{\mathtt{IPW}\}, \\
& \mathbb{E}[\hat\tau_l^{\mathtt{HON}}(\mathbf{x};K)] = \tau - \tau \mathbb{P}(n_0(\mathsf{t}) = 0 \text{ or } n_1(\mathsf{t}) = 0), \qquad l \in \{\mathtt{DIM}, \mathtt{SSE}\}.
\end{align*}
\end{lemma}
\section{Proofs}
\subsection{Proof of Lemma 4}\label{sa-sec: proof of multivarite darling erdos}
Taking $T = c \log(n)$ in \cite[Lemma 2.1]{horvath1993maximum}, we have
\begin{align*}
\lim_{n \rightarrow \infty}\mathbb{P} \bigg(\sup_{0 \leq t \leq c \log (n)} N(t) \leq \frac{z + b_d(c\log (n))}{a(c \log(n))} \bigg) = \exp \Big(-e^{-z} \Big).
\end{align*}
Now we expand the term $\frac{z + b_d(c\log (n))}{a(c \log(n))}$. For notational simplicity, denote
\begin{align*}
L = \log\log n, \qquad A = \log c, \qquad
L \to \infty \ \ (n \to \infty).
\end{align*}
First, we present some elementary expansions,
\[
\begin{aligned}
\sqrt{2(A+L)}
&= \sqrt{2L}\,\sqrt{1+\frac{A}{L}}
= \sqrt{2L}\Bigl(1+\frac{A}{2L}-\frac{A^{2}}{8L^{2}}
+\operatorname{O}\!\bigl(L^{-3}\bigr)\Bigr), \\
\frac{1}{\sqrt{2(A+L)}}
&= \frac{1}{\sqrt{2L}}
\Bigl(1-\frac{A}{2L}+\frac{3A^{2}}{8L^{2}}
+\operatorname{O}\!\bigl(L^{-3}\bigr)\Bigr), \\
\log(L+A)
&= \log L + \frac{A}{L} - \frac{A^{2}}{2L^{2}} + \operatorname{O}\!\bigl(L^{-3}\bigr).
\end{aligned}
\]
Now we expand the terms for the numerator $b_d(c \log(n))$,
\[
\begin{aligned}
N_{1} &= z + 2A + 2L + \frac{d}{2}\log \bigl(\log(c\log n)\bigr)
- \log \Gamma(d/2),\\
N_{2} &= z + 2A + 2L + \frac{d}{2}\log L - \log \Gamma(d/2),\\
N_{3} &= z + A + 2L + \frac{d}{2}\log L - \log \Gamma(d/2).
\end{aligned}
\]
Then
\begin{align*}
& \frac{z + b_d(c \log n)}{a(c \log(n))} - \frac{z + \log(c) + b_d(\log n)}{a(\log(n))} \\
& = \frac{N_1}{\sqrt{2(A+L)}} - \frac{N_3}{\sqrt{2L}} \\
& = N_1 \bigg(\frac{1}{\sqrt{2(A + L)}} - \frac{1}{\sqrt{2 L}}\bigg) + \frac{1}{\sqrt{2 L}}(N_1 - N_3) \\
& = N_1 \frac{1}{\sqrt{2 L}} \bigg(- \frac{A}{2 L} + \frac{3 A^2}{8 L^2} + O(L^{-3}) \bigg) + \frac{1}{\sqrt{2 L}} \bigg(\frac{d}{2} \Big(\frac{A}{L} - \frac{A^2}{2 L^2} + O(L^{-3})\Big) + A\bigg).
\end{align*}
Since $N_1 = 2L + O(\log \log \log(n))$, we have
\begin{align*}
& \frac{z + b_d(c \log n)}{a(c \log(n))} - \frac{z + \log(c) + b_d(\log n)}{a(\log(n))} = \frac{3 A^2}{4 \sqrt{2} L^{3/2}} + \frac{d A}{2 \sqrt{2} L^{3/2}} + o(L^{-3/2}) = o(L^{-1/2}).
\end{align*}
Since $a(\log(n)) = \Theta(L^{1/2})$, we have
\begin{align*}
& \mathbb{P} \bigg(\sup_{0 \leq t \leq c \log (n)} N(t) \leq \frac{z + c \log(n) + b_d(\log (n))}{a(\log(n))} \bigg) \\
& = \mathbb{P} \bigg(\sup_{0 \leq t \leq c \log (n)} N(t) \leq \frac{z + o(1) + b_d(c\log (n))}{a(c \log(n))} \bigg) \\
& = \mathbb{P} \bigg(a(c \log(n)) \sup_{0 \leq t \leq c \log (n)} N(t) - b_d(c\log (n)) \leq z + o(1)\bigg) \rightarrow \exp(-e^{-z}) \text{ as } n \rightarrow \infty,
\end{align*}
where the last line follows from convergence in distribution of $a(c \log(n)) \sup_{0 \leq t \leq c \log (n)} N(t) - b_d(c\log (n))$ to a continuous distribution and Slutsky's Theorem.
\subsection{Proof of Theorem~\ref{sa-thm:master}}
First, we introduce some notations. Recall for $\ell \in [p]$, $\pi_\ell$ denotes the permutation such that $(x_{\pi_\ell(i)}: 1 \leq i \leq n)$ is non-decreasing. Define sample mean at the left and right leave at index $k \in [n]$ based on coordinate $\ell \in [p]$ by
\begin{align*}
\hat\mu_{L}(k,\ell) = \frac{1}{k} \sum_{i = 1}^k y_{\pi_{\ell}(i)}, \qquad \hat \mu_{R}(k, \ell) = \frac{1}{n - k} \sum_{i = k+1}^n y_{\pi_{\ell}(i)}, \qquad k \in [n], \quad \ell \in [p].
\end{align*}
We can check that minimizing the \emph{sum of squares} criterion Equation~\eqref{eq:sse} is equivalent to maximizing the split criterion
\begin{align*}
(\hat \imath, \hat \jmath) = \operatorname*{arg\,max}_{(i,j) \in [n] \times [p]} \mathscr{I}(i,j).
\end{align*}
where
\begin{align*}
\mathscr{I}(k,\ell) & = \frac{k(n-k)}{n}\Big(\hat\mu_{L}(k,\ell)-\hat\mu_{R}(k,\ell)\Big)^2, \qquad k \in [n], \quad \ell \in [p].
\end{align*}
Moreover, under the constant conditional mean assumption, Assumption~\ref{sa-ass:DGP} (1), we have that $\hat\mu_{L}(k,\ell) - \hat \mu_{R}(k, \ell) = \frac{1}{k} \sum_{i = 1}^k \varepsilon_{\pi_{\ell}(i)} - \frac{1}{n - k} \sum_{i = k+1}^n \varepsilon_{\pi_{\ell}(i)}$. Hence we can w.l.o.g. replace $y_i$ by $\varepsilon_i$ in the definition of $\hat \mu_L$ and $\hat \mu_R$, that is,
\begin{align*}
\hat\mu_{L}(k,\ell) = \frac{1}{k} \sum_{i = 1}^k \varepsilon_{\pi_{\ell}(i)}, \qquad \hat \mu_{R}(k, \ell) = \frac{1}{n - k} \sum_{i = k+1}^n \varepsilon_{\pi_{\ell}(i)}, \qquad k \in [n], \quad \ell \in [p].
\end{align*}
The rest of the proof is organized as follows. In Section~\ref{sec:thm-master-uni}, we prove the results under $p = 1$, showing a strong approximation of the split criterion $(\mathscr{I}(k,1): k \in [n])$ by the square of a time-transformed Ornstein-Uhlenbeck (O-U) process, and studying the argmax of the split criterion through the argmax of the O-U process. In Section~\ref{sec:thm-master-multi}, we generalize to allow for $p \geq 1$. We show that the split criterion over different coordinates, that is, $(\mathscr{I}(k,\ell): k \in [n])$ for different $\ell$'s, are asymptotically independent. This reduces our problem to one-dimensional calculations, and the same technique of approximation by O-U process from Section~\ref{sec:thm-master-uni} can be used.
\subsubsection{Univariate Case} \label{sec:thm-master-uni}
This the case when $p = 1$. For notational simplicity, define partial sums by
\begin{align*}
S_k = \sum_{i = 1}^k \varepsilon_{\pi_1(i)}, \qquad k \in [n].
\end{align*}
By \citet[Equation A.4.37]{csorgo1997limit}, we can define a sequence of Brownian bridges $ \{ B_n(t) : 0 \leq t \leq 1 \} $ on a suitable probability space such that
\begin{equation} \label{eq:error_full}
\bigg|\max_{1 \leq k < n} \sqrt{\mathscr{I}(k,1)} - \sup_{1/n \leq t \leq 1-1/n}\frac{|B_n(t)|}{\sqrt{t(1-t)}} \bigg| =
\bigg|\max_{1 \leq k < n} \frac{\Big|\frac{1}{\sqrt{n}}S_k - \frac{k}{n}\frac{1}{\sqrt{n}}S_n \Big|}{\sqrt{(k/n)(1-k/n)}} - \sup_{1/n \leq t \leq 1-1/n}\frac{|B_n(t)|}{\sqrt{t(1-t)}} \bigg| = \epsilon_n,
\end{equation}
where $ \epsilon_n = o_{\mathbb{P}}\big((\log\log(n))^{-1/2}\big) $.
We note that while \citet[Equation A.4.37]{csorgo1997limit} bounds the approximation error of the maximum over the full range $ 1 \leq k < n $ as in \eqref{eq:error_full}, its proof, which relies on invariance principles for partial sums of i.i.d. random variables, can be generalized to bound the approximation error over $ 1 \leq k < n^{a} $, $ n^{b} < k < n $. Thus,
\begin{equation} \label{eq:error_partial}
\bigg|\max_{1 \leq k < n^{a}, \; n^{b} < k < n} \frac{\Big|\frac{1}{\sqrt{n}}S_k - \frac{k}{n}\frac{1}{\sqrt{n}}S_n \Big|}{\sqrt{(k/n)(1-k/n)}}- \sup_{1/n \leq t < n^{a-1}, \; n^{b-1} < t \leq 1-1/n}\frac{|B_n(t)|}{\sqrt{t(1-t)}} \bigg| = \epsilon_n.
\end{equation}
We note that the standardized Brownian bridge $ \big\{ B_n(t)/\sqrt{t(1-t)} : 0 < t < 1 \big\} $ is distributionally equivalent to a time-transformed Ornstein-Uhlenbeck (O-U) process $ \big\{ U(\log(t/(1-t))) : 0 < t < 1\big\} $, where $ \big\{U(t): t \in \mathbb{R}\big\}$ is an O-U process with mean $ \mathbb{E}[U(t)] =0 $ and covariance $\mathbb{E}[U(s)U(t)] = e^{-|s-t|/2}$ \citep[Section 1.9]{csorgo1981strong}, and thus
\begin{align}\label{eq:prob_orn}
\nonumber & \mathbb{P}\bigg( \sup_{ 1/n \leq t \leq 1-1/n} \frac{|B_n(t)|}{\sqrt{t(1-t)}} > \sup_{1/n \leq t < n^{a-1}, \; n^{b-1} < t \leq 1-1/n} \frac{|B_n(t)|}{\sqrt{t(1-t)}} + 2\epsilon_n\bigg) \\
\nonumber & = \mathbb{P}\bigg( \sup_{ -\log(n-1) \leq t \leq \log(n-1)} |U(t)| > \sup_{-\log(n-1) \leq t < \log(\frac{n^{a-1}}{1 - n^{a-1}}), \; \log(\frac{n^{b-1}}{1-n^{b-1}}) < t \leq \log(n-1)} |U(t)| + 2\epsilon_n\bigg) \\
& = \mathbb{P}\bigg( \sup_{ 0 \leq t \leq 2\log(n-1)} |U(t)| > \sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| + 2\epsilon_n \bigg),
\end{align}
where the last equality follows from stationarity of the process $ |U(t)| $, the square of which is a Cox-Ingersoll-Ross (CIR) process \citep{jaeschke2003survey}. Continuing from \eqref{eq:prob_orn}, for any sequence $u_n$, we have
\begin{equation}
\begin{aligned} \label{eq:prob_lower}
& \mathbb{P}\bigg( \sup_{ 0 \leq t \leq 2\log(n-1)} |U(t)| > \sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| + 2\epsilon_n \bigg) \\ & \quad \geq \mathbb{P}\bigg( \sup_{ 0 \leq t \leq 2\log(n-1)} |U(t)| \geq u_n, \sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| < u_n - 2\epsilon_n \bigg) \\ &
\quad \geq
\mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| < u_n - 2\epsilon_n \bigg) \\ & \hspace{9.5cm} - \mathbb{P}\bigg( \sup_{ 0 \leq t \leq 2\log(n-1)} |U(t)| < u_n \bigg).
\end{aligned}
\end{equation}
Now, since $ U(t) $ is a continuous, mean-zero Gaussian process, it induces a centered Gaussian measure on the space of all continuous functions on $\big[0,\,2\log(n-1)\big] $ equipped with the supremum norm (a separable Banach space). Thus, by the Gaussian correlation inequality \citep[Remark 3 (i)]{latala2017royen}, we have that
\begin{align} \label{eq:prob_lower_prod}
\nonumber & \mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| < u_n - 2\epsilon_n \bigg) \\
\nonumber &\quad \geq
\mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}})} |U(t)| < u_n - 2\epsilon_n \bigg)\cdot\mathbb{P}\bigg(\sup_{\log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| < u_n - 2\epsilon_n \bigg) \\
& \quad =
\mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}})} |U(t)| < u_n - 2\epsilon_n \bigg)\cdot\mathbb{P}\bigg(\sup_{0 < t \leq \log(\frac{n^{1-b}(n-1)}{1-n^{b-1}})
} |U(t)| < u_n - 2\epsilon_n \bigg),
\end{align}
where the last equality follows from stationarity.
\begin{remark}\label{remark: Eicker (1979) error}
The next step of our proof relies on a precise characterization of weak convergence for the suprema of a standardized empirical process, as studied in \citep{eicker1979asymptotic}. However,
\citet[Theorem 5]{eicker1979asymptotic} is incorrectly stated: the $2\log(c)$ term appearing in the limiting probability should be $\log(c)$. This correction has important implications in our proof.
\end{remark}
By the Darling-Erd\H{o}s Limit Theorem for the O-U process \citep[Theorem 1.9.1]{csorgo1981strong} and \citep[Theorem 2.2 and the correct version of Theorem 5]{eicker1979asymptotic}, for all $ c > 0 $ and $ z \in \mathbb{R} $, we have
\begin{align}\label{eq:darling-erdos}
\nonumber & \lim_{n\to\infty} \mathbb{P}\bigg(\sup_{ 0 \leq t \leq (c+o(1))\log(n)}|U(t)| < \frac{2\log\log(n) + (1/2)\log\log\log(n) + z - (1/2)\log(\pi)}{\sqrt{2\log\log(n)}} \bigg) \\
& \qquad\qquad\qquad = \exp\Big(-e^{-(z-\log(c))}\Big).
\end{align}
For a detailed proof of a generalized result on multidimensional O-U process, see Section~\ref{sa-sec: proof of multivarite darling erdos}.
Let $ z^* $ maximize $ z \mapsto \exp\big(-2e^{-(z-\log(2-(b-a)))}\big) - \exp\big(-2e^{-(z-\log(2))}\big) $,
and set $$ u_n = \frac{2\log\log(n) + (1/2)\log\log\log(n) + z^* - (1/2)\log(\pi)}{\sqrt{2\log\log(n)}}. $$
We combine \eqref{eq:prob_orn}, \eqref{eq:prob_lower}, and \eqref{eq:prob_lower_prod}, and employ \eqref{eq:darling-erdos} three times with $ c = 2 $ and $ c = 2-b $, and $ c = a $, together with the fact that $ \epsilon_n = o_{\mathbb{P}}\big((\log\log(n))^{-1/2}\big) $. We have that
\begin{align}
\nonumber & \liminf_{n\rightarrow\infty}\mathbb{P}\bigg( \sup_{ 1/n \leq t \leq 1-1/n} \frac{|B_n(t)|}{\sqrt{t(1-t)}} > \sup_{1/n \leq t < n^{a-1}, \; n^{b-1} < t \leq 1-1/n} \frac{|B_n(t)|}{\sqrt{t(1-t)}} + 2\epsilon_n\bigg) \\
\nonumber & \qquad \geq \exp\Big(-2e^{-(z^*-\log(a))}\Big) \cdot \exp\Big(-2e^{-(z^*-\log(2-b))}\Big) - \exp\Big(-2e^{-(z^*-\log(2))}\Big)
\\ & \qquad =
\nonumber \exp\Big(-2e^{-(z^*-\log(2-(b-a)))}\Big) - \exp\Big(-2e^{-(z^*-\log(2))}\Big) \\
\nonumber & \qquad = \frac{b-a}{2}\bigg(1-\frac{b-a}{2}\bigg)^{\frac{2}{b-a}-1}
\\ & \qquad \geq \frac{b-a}{2e}.
\end{align}
\begin{remark}
Alternatively, for any $ 0 < A < B < C $, we have
\begin{equation} \label{eq:max-ou}
\mathbb{P}\bigg( \sup_{0 \leq t \leq C} |U(t)| > \sup_{0 \leq t \leq A, \; B \leq t \leq C} |U(t)| \bigg) = \frac{B-A}{C}.
\end{equation}
This can readily be shown using the fact that the absolute value of a zero-mean O-U process is stationary, Markov, and has continuous paths. Consequently, ignoring the stochastic error $\epsilon_n$ from approximating the split criterion \eqref{eq:Delta} by the square of a standardized Brownian bridge (not yet justified), using \eqref{eq:max-ou}, we can approximate the probability $\mathbb{P} (\max_{1 \leq k \leq n} \mathscr{I}(k,1) > \max_{1 \leq k < n^a, n^b < k < n} \mathscr{I}(k,1))$ by
\begin{align}
\nonumber & \mathbb{P}\bigg( \sup_{ 0 \leq t \leq 2\log(n-1)} |U(t)| > \sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |U(t)| \bigg)
\\ & \qquad = \frac{\log\frac{n^{b-1}(n-1)}{1-n^{b-1}})-\log(\frac{n^{a-1}(n-1)}{1-n^{a-1}})}{2\log(n-1)} \rightarrow \frac{b-a}{2}, \quad n \rightarrow \infty.
\end{align}
\end{remark}
\subsubsection{Multivariate Case}\label{sec:thm-master-multi}
Now we prove for the general case of $p \geq 1$. As a sketch of the proof, we show that the split criterion over different coordinates, that is, $(\mathscr{I}(k,\ell): k \in [n])$ for different $\ell$'s, are asymptotically independent, which will imply
\begin{align*}
& \liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b}, \hat \jmath = \ell \big) \\
& = \liminf_n\mathbb{P}\Big(\max_k\mathscr{I}(k,1) > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1) \Big) \\ & \geq
\liminf_n\mathbb{P}\Big(\max_k\mathscr{I}(k,1) > z_n > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k}\mathscr{I}(k,1) > z_n > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1) \Big) \\ & \stackrel{(*)}{=} \Big(\liminf_n\mathbb{P}\Big(\max_{k}\mathscr{I}(k,1) < z_n\Big)\Big)^{p-1}\liminf_n\mathbb{P}\Big(\max_{k}\mathscr{I}(k,1) > z_n > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1) \Big).
\end{align*}
where in equality (*) we use asymptotic independence between $\mathscr{I}(\cdot,\ell)$ for different $\ell$'s, and the last line are one-dimensional probabilities that can be handled by O-U process approximation like in Section~\ref{sec:thm-master-uni}.
To show the split criteria over different coordinates are asymptotically independent, we break down into two steps: In the first step, we show the partial sum process $n$ indices and $p$ coordinates can be approximated by another partial sum process with Gaussian increments (hence a Gaussian process), with the same covariance structure. In the second step, we show the covariance between the split criteria over any two different coordinates and any indices are vanishing. Together with Gaussianity, this implies that the split criteria over different coordinates are asymptotically independent.
\begin{center}
\textbf{Step 1: Non-Gaussian to Gaussian Coupling.}
\end{center}
For $1 \leq \ell \leq p$, denote by $H_n^\ell(\frac{k}{n})$ the scaled partial sum for the $\ell$-th coordinate evaluated at \textit{time} $\frac{k}{n}$, that is,
\begin{align*}
H_n^\ell\bigg(\frac{k}{n}\bigg) & = \sqrt{\frac{n}{k(n-k)}} \bigg\{\sum_{i = 1}^k \varepsilon_{\pi^\ell(i)} - \frac{k}{n} \sum_{i = 1}^n \varepsilon_{\pi^\ell(i)} \bigg\} \\
& = \sqrt{\frac{n}{k(n-k)}} \sum_{i=1}^n \Big(\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n}\Big) \varepsilon_i,
\end{align*}
where $\# \pi^{\ell}:[n] \rightarrow [n]$ is the inverse mapping of $\pi^{\ell}$.
We use a truncation argument for the proof. Fix $\varepsilon \in (0,1)$. Take $r_n = \exp((\log n)^{\varepsilon})$. And consider
\begin{align*}
\mathbf{C}_i = \sqrt{n}\Big(\Big(\sqrt{\frac{n}{k(n-k)}} (\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n}): r_n \leq k \leq n - r_n\Big)^\top: 1 \leq \ell \leq p\Big)^\top \varepsilon_i,
\end{align*}
where $\# \pi^{\ell}$ denotes the inverse mapping of $\pi^{\ell}$. Notice that we add the $\sqrt{n}$ factor for standardization. Then we can check that condition on $\mathscr{B}$, the $\sigma$-algebra generateed by the $p$ permulations $\pi^1, \cdots, \pi^p$, $\mathbf{C}_i$'s are independent, and for all $1 \leq j \leq p(n - 2 r_n)$, $1 \leq \ell \leq p$, we have
\begin{align*}
n^{-1}\sum_{i = 1}^n \mathbb{E}[C_{ij}^2|\mathscr{B}]
& = \frac{n}{k (n - k)} \bigg[k \Big(\frac{n - k}{n}\Big)^2 + (n - k) \Big(\frac{k}{n}\Big)^2 \bigg] = 1,
\end{align*}
where we assume row $j$ in $\mathbf{C}_i$ corresponds to $\sqrt{n} \sqrt{\frac{n}{k(n-k)}} (\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n})$.
To use the coupling result \cite[Theorem 2.1]{chernozhukov2017central}, we bound a few quantities: Suppose $K_1$ and $K_2$ are the universal constants given in the cited theorem,
\begin{align}\label{eq:third moment coupling}
\nonumber L_n & = \max_{1 \leq j \leq p (n - 2 r_n)} \sum_{i = 1}^n \mathbb{E}[|C_{ij}|^3|\mathscr{B}]/n \\
\nonumber & = \max_{1 \leq \ell \leq p} \max_{r_n \leq k \leq n - r_n} n^{3/2} \Big(\frac{n}{k (n - k)}\Big)^{3/2} \Big[k (1 - k/n)^3 + (n - k)(-k/n)^3 \Big] \mathbb{E}[|\varepsilon_i|^3] /n \\
\nonumber & \lesssim \max_{1 \leq \ell \leq p} \max_{r_n \leq k \leq n - r_n} \frac{(n - 2k) \sqrt{n}}{\sqrt{(n - k) n k}} \\
& \lesssim \sqrt{n/r_n}.
\end{align}
For notational simplicity, denote $\mathtt{P} = p (n - 2 r_n)$. Take $\bar{L}_n = L_n$, then
\begin{align*}
\phi_n = K_2 \bigg(\frac{\bar{L}_n^2 \log^4(\mathtt{P})}{n}\bigg)^{-1/6}
= K_2 \bigg(\frac{r_n}{\log^4(\mathtt{P})}\bigg)^{1/6}.
\end{align*}
The definition of $\mathbf{C}_i$ implies $C_{ij}$ is $\sqrt{n/r_n}$-exponential. Hence
\begin{align*}
M_{n,X}(\phi_n) & = n^{-1} \sum_{i = 1}^n \mathbb{E} \bigg[\max_{1 \leq j \leq \mathtt{P}} |C_{ij}|^3 \mathds{1}\Big(\max_{1 \leq j \leq \mathtt{P}} |C_{ij}| > \sqrt{n}/(4 \phi_n \log(\mathtt{P}))\Big) \bigg| \mathscr{B}\bigg] \\
& \leq n^{-1} \sum_{i = 1}^n \mathbb{E} \bigg[\max_{1 \leq j \leq \mathtt{P}} |C_{ij}|^6 \bigg| \mathscr{B} \bigg]^{1/2} \mathbb{P} \bigg[\max_{1 \leq j \leq \mathtt{P}} |C_{ij}| > \sqrt{n}/(4 \phi_n \log(\mathtt{P}))\bigg| \mathscr{B} \bigg]^{1/2} \\
& \leq n^{-1} \sum_{i = 1}^n \bigg[\sum_{1 \leq j \leq \mathtt{P}} \mathbb{E}[C_{ij}^6| \mathscr{B}] \bigg]^{1/2} \bigg[\sum_{1 \leq j \leq \mathtt{P}} \mathbb{P} \Big(|C_{ij}| > \sqrt{n}/(4 \phi_n \log(\mathtt{P}))\Big| \mathscr{B}\Big)\bigg]^{1/2} \\
& \lesssim n^{-1} \sum_{i = 1}^n (\mathtt{P} (n/r_n)^3)^{1/2} \bigg[\mathtt{P} \exp \Big( -\frac{\sqrt{n}/(4 \phi_n \log(\mathtt{P}))}{\sqrt{n/r_n}}\Big) \bigg]^{1/2} \\
& \lesssim \mathtt{P} (n/r_n^3)^{1/2} \exp \Big(- \frac{1}{4}\Big(\frac{r_n}{\log \mathtt{P}}\Big)^{1/3} \Big) \\
& \lesssim n^{-2},
\end{align*}
since $r_n = \exp((\log n)^{\varepsilon})$ and $\varepsilon, p$ are fixed. Now condition on $\mathscr{B}$, let $\boldsymbol{D}_i, 1 \leq i \leq n$ to be independent mean-zero Gaussian random vectors such that
\begin{align*}
\boldsymbol{D}_i \sim N(\mathbf{0}, \mathbb{E}[\mathbf{C}_i \mathbf{C}_i^{\top}|\mathscr{B}]), \qquad \text{condition on } \mathscr{B}.
\end{align*}
Then for each $1 \leq j \leq \mathtt{P}$, $1 \leq i \leq n$, we have $D_{ij}$ is $r_n^{-1}$-subGaussian. Hence the same argument implies
\begin{align*}
M_{n,Y}(\phi_n) \lesssim n^{-2}.
\end{align*}
\cite[Theorem 2.1]{chernozhukov2017central} then implies
\begin{align}\label{eq: clt retangles}
\nonumber \sup_{A \in \mathcal{A}^{re}} \bigg|\mathbb{P} \Big(\sum_{i= 1}^n \mathbf{C}_i \in A \Big| \mathscr{B} \Big) - \mathbb{P} \Big(\sum_{i=1}^n \boldsymbol{D}_i \in A \Big| \mathscr{B} \Big)\bigg|
& \leq K_1 \bigg[\Big(\frac{\bar{L}_n^2 \log^7(\mathtt{P})}{n}\Big)^{1/6}
+ \frac{M_{n,X}(\phi_n) + M_{n,Y}(\phi_n)}{\bar{L}_n}\bigg] \\
\nonumber & \lesssim \bigg(\frac{\log^7(\mathtt{P})}{r_n}\bigg)^{1/6} + \sqrt{\frac{r_n}{n}} \frac{1}{n^2} \\
& \lesssim \bigg(\frac{\log^7(n)}{r_n}\bigg)^{1/6},
\end{align}
where $\mathcal{A}^\text{re}$ is the class of all rectangles $A$ of the form
\begin{align*}
A = \{\mathbf{u} \in \mathbb{R}^{\mathtt{P}}: a_j \leq u_j \leq b_j, \forall j = 1, 2, \cdots, \mathtt{P}\},
\end{align*}
for some $- \infty \leq a_j \leq b_j \leq \infty$, $j = 1,2, \cdots, \mathtt{P}$. In particular, suppose $u_i, 1 \leq i \leq n$ are i.i.d $N(0, \mathbb{E}[\varepsilon_i^2])$ random variables, then $\boldsymbol{D}_i$ can be taken such that
\begin{align*}
\boldsymbol{D}_i = \sqrt{n}\Big(\Big(\sqrt{\frac{n}{k(n-k)}} (\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n}): r_n \leq k \leq n - r_n\Big)^\top: 1 \leq \ell \leq p\Big)^\top u_i.
\end{align*}
The above result shows if we define
\begin{align*}
G_n^\ell\bigg(\frac{k}{n}\bigg) = \sqrt{\frac{n}{k(n-k)}} \bigg\{\sum_{i = 1}^k u_{\pi^\ell(i)} - \frac{k}{n} \sum_{i = 1}^n u_{\pi^\ell(i)} \bigg\},
\end{align*}
then Equation~\eqref{eq: clt retangles} and unconditioning on $\mathscr{B}$, we get
\begin{align*}
\sup_{t_1, \cdots t_p \in \mathbb{R}}\Big|\mathbb{P}\Big(\max_{r_n \leq k \leq n - r_n} |H_n^\ell(k/n)| \leq t_\ell, 1 \leq \ell \leq p\Big) - \mathbb{P}\Big(\max_{r_n \leq k \leq n - r_n}
|G_n^\ell(k/n)| \leq t_\ell, 1 \leq \ell \leq p\Big)\Big|
\lesssim \bigg(\frac{\log^7(n)}{r_n}\bigg)^{1/6}.
\end{align*}
\begin{center}
\textbf{Step 2: Gaussian to Gaussian Coupling.}
\end{center}
For $1 \leq \ell \leq p$, denote by $G_n^\ell(\frac{k}{n})$ the partial sum for the $\ell$-th coordinate evaluated at \textit{time} $\frac{k}{n}$, that is,
\begin{align*}
G_n^\ell\bigg(\frac{k}{n}\bigg) = \sqrt{\frac{n}{k(n-k)}} \bigg\{\sum_{i = 1}^k u_{\pi^\ell(i)} - \frac{k}{n} \sum_{i = 1}^n u_{\pi^\ell(i)} \bigg\}.
\end{align*}
Then $\mathbf{G}_n = ((G_n^1(1/n),G_n^1(2/n),\cdots,G_n^1(n/n))^\top, \cdots, (G_n^p(1/n),G_n^p(2/n),\cdots,G_n^p(n/n))^\top)^\top$. Then $\mathbf{G}_n$ is a $np$-dimensional Gaussian random vector, and denote by $\boldsymbol{\Sigma}_n$ its covariance matrix. We want to show that $\boldsymbol{\Sigma}_n$ is close to one with covariance between different coordinates zero.
Consider two different coordinates, $\ell_1, \ell_2 \in [p]$. W.l.o.g, we can assume $\ell_1 = 1$ and $\ell_2 = 2$. Let $k,j \in [n]$. Denote by $\boldsymbol{\sigma}$ the sigma-algebra generated by $\pi_1, \cdots, \pi_p$. Then
\begin{align*}
& \operatorname{Cov} \bigg[G_n^1\bigg(\frac{k}{n}\bigg), G_n^2\bigg(\frac{j}{n}\bigg)\bigg| \boldsymbol{\sigma} \bigg] \\
= & \sqrt{\frac{n}{k(n-k)} \frac{n}{j(n-j)}} \bigg\{\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] - \frac{j}{n} \sum_{i = 1}^k \sum_{i^{\prime} = 1}^n \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] \\
& \qquad \qquad \qquad \qquad - \frac{k}{n}\sum_{i = 1}^n\sum_{i^{\prime} = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] + \frac{kj}{n^2} \sum_{i =1}^n \sum_{i^{\prime} = 1}^n \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}]\bigg\} \\
= & \sqrt{\frac{n}{k(n-k)} \frac{n}{j(n-j)}} \frac{jk}{n}\bigg\{\frac{n}{jk}\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] - 1\bigg\}.
\end{align*}
To calculate $\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}]$, we can first condition on $\pi_1$, and let $\mathcal{I} = \{\pi_1(i): 1 \leq i \leq k\}$. Observe that $\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] = |\{i^{\prime} \in [j]: \pi_1(i^{\prime}) \in \mathcal{I}\}|$. Now consider $$f(\pi) = \frac{n}{jk}|\{i \in [j]: \pi(i) \in \mathcal{I}\}|,$$ $\pi$ is a random permutation of $[n]$. Changing the order of the first $j$ values of $\pi$ does not change the value of $f(\pi)$, and $|f(\pi) - f(\pi^{s,t})| \leq \frac{n}{jk}$ for all $\pi$, $s \in \{1,\cdots,j\}$, $t \in \{j+1,\cdots,n\}$, where the permutation $\pi^{s,t}$ is obtained from $\pi$ by transposition of its $s$th and $t$th coordinates. We will show later that w.l.o.g. we can assume $j,k \leq \lceil n/2 \rceil$. Then by Lemma 2 from \cite{el2009transductive}, for any $t \geq 0$,
\begin{align*}
& \mathbb{P}\bigg(\bigg|\frac{n}{jk}\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] - 1 \bigg| \geq t \bigg|\pi_1\bigg) \\
= & \mathbb{P}(|f(\pi_2) - \mathbb{E}[f(\pi_2)]| \geq t|\pi_1) \\
\leq & 2 \exp \bigg(- \frac{2 t^2}{j (\frac{n}{jk})^2} \frac{n - 1/2}{n - j}(1 - \frac{1}{2 \max(j,n-j)})\bigg).
\end{align*}
Since $\frac{n - 1/2}{n - j}(1 - \frac{1}{2 \max(j,n-j)}) \geq 1 - \frac{1}{n}$, we can marginalize over $\pi_1$ and uncondition on $\boldsymbol{\sigma}$ to get there exists a positive constant $C$ such that for $n$ large enough, for all $j,k \in [n]$,
\begin{align*}
|\frac{n}{jk}\sum_{i = 1}^k \sum_{i^\prime = 1}^j \mathbb{E}[u_{\pi_1(i)} u_{\pi_2(i^{\prime})}|\boldsymbol{\sigma}] - 1|
\leq C\frac{n}{\sqrt{j}k}.
\end{align*}
which implies
\begin{align}\label{eq: cov}
|\operatorname{Cov} [G_n^1(\frac{k}{n}), G_n^2(\frac{j}{n})]| \leq C \sqrt{\frac{jk}{(n-k)(n-j)}} \frac{n}{\sqrt{jk}} \frac{1}{\sqrt{k}}
\leq C k^{-1/2}.
\end{align}
The reduction to $j,k \leq \lceil n/2 \rceil$ is because
\begin{align*}
G_n^\ell\bigg(\frac{k}{n}\bigg) & = \sqrt{\frac{n}{k(n-k)}} \bigg\{\sum_{i = 1}^k u_{\pi^\ell(i)} - \frac{k}{n} \sum_{i = 1}^n u_{\pi^\ell(i)} \bigg\} \\
& = - \sqrt{\frac{n}{k(n-k)}} \bigg\{\sum_{i = k+1}^n u_{\pi^\ell(i)} - \frac{n-k}{n} \sum_{i = 1}^n u_{\pi^\ell(i)} \bigg\}.
\end{align*}
Now consider a $np$-dimensional mean-zero Gaussian random vector $$\mathbf{Z}_n = ((Z_n^1(1/n),Z_n^1(2/n),\cdots,Z_n^1(n/n))^\top, \cdots, (Z_n^p(1/n),Z_n^p(2/n),\cdots,Z_n^p(n/n))^\top)^\top,$$ where for each $1 \leq \ell \leq p$, $(Z_n^\ell(1/n),Z_n^\ell(2/n),\cdots,Z_n^\ell(n/n))^\top$ has the same joint distribution as the partial sum random vector $(G_n^\ell(1/n)$,$G_n^\ell(2/n),\cdots,G_n^\ell(n/n))^\top$, and for any $\ell \neq \ell^{\prime}$ and any $j,k \in [n]$,
\begin{align*}
\operatorname{Cov}[Z_n^\ell(j/n),Z_n^{\ell^{\prime}}(k/n)] = 0.
\end{align*}
Denote by $\boldsymbol{\Gamma}_n$ the covariance matrix of $\mathbf{Z}_n$. We want to show $\boldsymbol{\Gamma}_n$ is close to $\boldsymbol{\Sigma}_n$. For a tight control on the rate of convergence, consider the truncated random vector,
\begin{align*}
T_{r_n}(\mathbf{G}_n) = ((G_n^\ell(k/n): r_n \leq k \leq n - r_n)^\top: 1 \leq \ell \leq p)^\top, \\
T_{r_n}(\mathbf{Z}_n) = ((Z_n^\ell(k/n): r_n \leq k \leq n - r_n)^\top: 1 \leq \ell \leq p)^\top.
\end{align*}
Also by an abuse of notations, denote by $T_{r_n}(\boldsymbol{\Sigma}_n)$ and $T_{r_n}(\boldsymbol{\Gamma}_n)$ the covariance matrix of $T_{r_n}(\mathbf{G}_n)$ and $T_{r_n}(\mathbf{Z}_n)$, respectively. Then Equation~\eqref{eq: cov} implies
\begin{align}\label{eq: matrix comparison}
\lVert {T_{r_n}(\boldsymbol{\Sigma}_n) - T_{r_n}(\boldsymbol{\Gamma}_n)}\rVert_{\operatorname{max}} = O(r_n^{-1/2}).
\end{align}
Additionally, we can lower bound the variance of each item of $T_{r_n}(\mathbf{Z}_n)$ by the following conditioning argument: Condition on the permutations $\pi_\ell$, $1 \leq \ell \leq p$, then
\begin{align*}
\mathbb{V}[\mathbf{Z}_n^\ell(k/n)|\pi_\ell, 1 \leq \ell \leq p]
& = \mathbb{V}[\mathbf{G}_n^\ell(k/n)|\pi_\ell, 1 \leq \ell \leq p] \\
& = \mathbb{V}\bigg[\sqrt{\frac{n}{k(n - k)}} \Big(\sum_{i = 1}^k u_{\pi^\ell(i)} - \frac{k}{n} \sum_{i = 1}^n u_{\pi^\ell(i)}\Big)\bigg|\pi_\ell, 1 \leq \ell \leq p\bigg] \\
& = \mathbb{V}\bigg[\sqrt{\frac{n}{k(n - k)}} \Big(\sum_{i = 1}^k u_i - \frac{k}{n} \sum_{i = 1}^n u_i\Big)\bigg] \\
& = 1, \qquad 1 \leq k < n, 1 \leq \ell \leq p,
\end{align*}
where in the third line, we have used the fact that condition on $\pi_\ell, 1 \leq \ell \leq p$, $(u_{\pi^\ell(i)})_{i \in [n]}$'s are i.i.d. $\mathsf{N}(0,1)$. By the Gaussian-to-Gaussian Comparison result \cite[Proposition 2.1]{chernozhuokov2022improved},
\begin{align*}
\sup_{\mathbf{y} \in \mathbb{R}^{p T(n)}} |\mathbb{P}(T_{r_n}(\mathbf{G}_n) \leq \mathbf{y}) - \mathbb{P}(T_{r_n}(\mathbf{Z}_n) \leq \mathbf{y})| \leq C \log (n) \lVert {T_{r_n}(\boldsymbol{\Sigma}_n) - T_{r_n}(\boldsymbol{\Gamma}_n)}\rVert_{\operatorname{max}},
\end{align*}
where $C$ is an absolute constant, and $T(n) = \lceil n - r_n \rceil - \lfloor r_n \rfloor$. Combining with Equation~\eqref{eq: matrix comparison} and taking $\mathbf{y} = (t_1 \mathbf{1}^\top, \cdots t_p \mathbf{1}^\top)$, $\mathbf{y} = -(t_1 \mathbf{1}^\top, \cdots t_p \mathbf{1}^\top)$ separately with $\mathbf{1}$ a vector of $T(n)$ 1's, we get
\begin{align}\label{eq: approax}
\nonumber & \sup_{t_1, \cdots t_p \in \mathbb{R}}\Big|\mathbb{P}\Big(\max_{r_n \leq k \leq n - r_n} |G_n^\ell(k/n)| \leq t_\ell, 1 \leq \ell \leq p\Big) - \mathbb{P}\Big(\max_{r_n \leq k \leq n - r_n}
|Z_n^\ell(k/n)| \leq t_\ell, 1 \leq \ell \leq p\Big)\Big| \\
& = O(\log(n) r_n^{-1/2}).
\end{align}
\begin{center}
\textbf{Step 3: Reduction of calculations of one-dimensional O-U process}
\end{center}
As in the previous two sections, fix $\varepsilon > 0$, and take $r_n = \exp((\log n)^{\varepsilon})$. Let $\mathcal{E} = \{\exists \ell \in [p]: \operatorname*{arg\,max}_k \mathscr{I}(k,\ell) < r_n \text{ or } \operatorname*{arg\,max}_k \mathscr{I}(k,\ell) > n - r_n\}$. Then by \cite[proof of Theorem A.4.2]{csorgo1997limit}, and a union bound argument, we have
\begin{align*}
\mathbb{P}(\mathcal{E}) \leq \sum_{\ell = 1}^p \mathbb{P}(\operatorname*{arg\,max}_k \mathscr{I}(k,\ell) < r_n \text{ or } \operatorname*{arg\,max}_k \mathscr{I}(k,\ell) > n - r_n) = o(1).
\end{align*}
Hence we can effectively restrict the candidates of $\operatorname*{arg\,max}$ to $[r_n, n - r_n]$. W.l.o.g., we consider coordinate $\ell = 1$, and
\begin{align*}
& \liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b}, \hat \jmath = \ell \big) \\
& = \mathbb{P}\Big(\max_{k \in [n]}\mathscr{I}(k,1) > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k \in [n]}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1) \Big) \\
& \geq \mathbb{P}\Big(\max_{k \in [n]}\mathscr{I}(k,1) > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k \in [n]}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1),\mathcal{E}^c \Big) - \mathbb{P}(\mathcal{E})\\
& \geq \mathbb{P}\Big(\max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1)\Big) - 2 \mathbb{P}(\mathcal{E}) \\
& \geq \mathbb{P}\Big(\max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{k,j\neq 1}\mathscr{I}(k,j), \; \max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1)\Big) + o(1).
\end{align*}
Now we can using the \textit{coupling result} developed previously. Using our notation, we have $\mathscr{I}(k,\ell) = (H_n^\ell(k/n))^2$. Hence
\begin{align*}
& \mathbb{P}\Big(\max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{j\neq 1, k \in [r_n, n - r_n]}\mathscr{I}(k,j), \; \max_{k \in [r_n, n - r_n]}\mathscr{I}(k,1) > \max_{k \notin [n^a,n^b]}\mathscr{I}(k,1) \Big) \\
& = \mathbb{P} \Big(\max_{k \in [r_n, n - r_n]} |H_n^1\Big(\frac{k}{n}\Big)| > \max_{\ell \neq 1, k \in [r_n, n - r_n]}|G_n^\ell\Big(\frac{k}{n}\Big)|, \; \max_{k \in [r_n, n - r_n]} |H_n^1\Big(\frac{k}{n}\Big)| > \max_{k \notin [n^a,n^b]} |H_n^1\Big(\frac{k}{n}\Big)| \Big) \\
& \geq \sup_{z \in \mathbb{R}}\mathbb{P} \Big(\max_{k \in [r_n, n - r_n]} |H_n^1\Big(\frac{k}{n}\Big)| > z > \max_{\ell \neq 1, k \in [r_n, n - r_n]}|H_n^\ell\Big(\frac{k}{n}\Big)|, \; \max_{k \in [r_n, n - r_n]} |H_n^1\Big(\frac{k}{n}\Big)| > z > \max_{k \notin [n^a,n^b]} |H_n^1\Big(\frac{k}{n}\Big)| \Big) \\
& \geq \sup_{z \in \mathbb{R}}\mathbb{P} \Big(\max_{k \in [r_n, n - r_n]} |Z_n^1\Big(\frac{k}{n}\Big)| > z > \max_{\ell \neq 1, k \in [r_n, n - r_n]}|Z_n^\ell\Big(\frac{k}{n}\Big)|, \; \max_{k \in [r_n, n - r_n]} |Z_n^1\Big(\frac{k}{n}\Big)| > z > \max_{k \notin [n^a,n^b]} |Z_n^1\Big(\frac{k}{n}\Big)| \Big) \\
& \qquad + O(\log(n)^{7/6} r_n^{-1/6}),
\end{align*}
where we have used Lemma~\ref{lem: approximation -- balanced region} and Lemma~\ref{lem: approximation -- imbalanced region}. Since we choose $r_n = \exp((\log n)^{\varepsilon})$, we have $\log(n)^{7/6} r_n^{-1/6} = o(1)$. It then follows from independence and symmetry between $Z_n^\ell$'s across different $\ell$'s that
\begin{align*}
& \liminf_{n \rightarrow \infty} \sup_{z \in \mathbb{R}} \mathbb{P} \Big(\max_{k \in [r_n, n - r_n]} |Z_n^1\Big(\frac{k}{n}\Big)| > z > \max_{\ell \neq 1, k \in [r_n, n - r_n]}|Z_n^\ell\Big(\frac{k}{n}\Big)|, \; \max_{k \in [r_n, n - r_n]} |Z_n^1\Big(\frac{k}{n}\Big)| > z > \max_{k \notin [n^a,n^b]} |Z_n^1\Big(\frac{k}{n}\Big)| \Big) \\
& \geq \liminf_{n \rightarrow \infty} \sup_{z \in \mathbb{R}} \mathbb{P}(\max_{k \in [r_n, n - r_n]}|Z_n^1\Big(\frac{k}{n}\Big)| < z)^{p-1} \mathbb{P}(\max_{k \in [r_n, n - r_n]}|Z_n^1\Big(\frac{k}{n}\Big)| > z > \max_{k \in [r_n, n - r_n], k \notin [n^a, n^b]}|Z_n^1\Big(\frac{k}{n}\Big)| )\\
& \geq \sup_z \exp\Big(-2(p-1)e^{-(z-\log(2))}\Big)\Big( \exp\Big(-2e^{-(z-\log(2-(b-a)))}\Big) - \exp\Big(-2e^{-(z-\log(2))}\Big)\Big) \\
& = \frac{b-a}{2p}\bigg(1-\frac{b-a}{2p}\bigg)^{\frac{2p}{b-a}-1} \\ & \geq \frac{b-a}{2pe},
\end{align*}
where the third line is by similar calculation as in Section~\ref{sec:thm-master-uni}. Putting together, we have
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath \leq n^{b}, \hat \jmath = \ell \big) \geq \frac{b - a}{2 p e},
\end{align*}
and by symmetry, we have
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\big( n - n^{a} \leq \hat\imath \leq n -n^{b}, \hat \jmath = \ell \big) \geq \frac{b - a}{2 p e}.
\end{align*}
\subsection{Proof of Theorem~\ref{sa-thm:rates}}
For simplicity, we denote $\hat{\mu}^{\mathtt{NSS}}(\mathbf{x})$ by $\hat{\mu}(\mathbf{x})$. We divide the proofs into two parts, one for uniform estimation and one for pointwise results near the boundary.
\begin{center}
\textbf{Part 1: Inconsistency for Uniform Estimation Rates}
\end{center}
For notational simplicity, introduce the \emph{partial sum based on ordering for the $\ell$'s coordinate},
\begin{align*}
S(k,\ell) = \sum_{i = 1}^{k} \varepsilon_{\pi_{\ell}(i)}, \qquad k \in [n], \quad \ell \in [p],
\end{align*}
and define the optimal index for splitting based on the $\ell$'s coordinate by
\begin{align*}
\imath_{\ell} = \operatorname*{arg\,max}_{k \in [n]} \mathscr{I}(k,\ell), \qquad \ell \in [p].
\end{align*}
Consider the event
\begin{align*}
\mathtt{Imbalance}_{\ell} & = \{\hat \jmath = \ell, \hat \imath < n^b \text{ or } \hat \imath > n - n^b\} \\
& = \{\max_k\mathscr{I}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}(k,j), \; \max_{k}\mathscr{I}(k,\ell) > \max_{k \in [n^b, n - n^b]}\mathscr{I}(k,\ell)\}, \qquad \ell \in [p].
\end{align*}
Consider the case $\hat \imath < n^b$ on $\mathtt{Imbalance}_{\ell}$. The other case where $\hat \imath > n - n^b$ can be dealt with by symmetry. Then
\begin{align*}
& \sup_{\mathbf{x} \in \mathcal{X}} |\hat{\mu}(\mathbf{x}) - \mu|^2 \\
& \geq \frac{S(\imath_{\ell}, \ell)^2}{\imath_{\ell}^2} \\
& \geq \frac{1}{\imath_{\ell}} \bigg[\frac{S(\imath_{\ell}, \ell)^2}{\imath_{\ell}} + \frac{(S(n,\ell) - S(\imath_{\ell}, \ell))^2}{n - \imath_{\ell}} - \frac{(S(n,\ell) - S(\imath_{\ell}, \ell))^2}{n - \imath_{\ell}}\bigg] \\
& \geq \frac{1}{\min\{\imath_{\ell}, n - \imath_{\ell}\}} \bigg(\max_{k \in [n]} \Big(\frac{S(k, \ell)^2}{k} + \frac{(n - S(k, \ell))^2}{n - k} \Big) - \max_{\lfloor n/2 \rfloor \leq k \leq n}\frac{S(k, \ell)^2}{k} - \max_{1 \leq k \leq \lceil n/2 \rceil}\frac{(n - S(k, \ell))^2}{n - k} \bigg).
\end{align*}
where the last line is because $\imath_{\ell}$ is the index that maximize the split criterion based on the $\ell$'s coordinate, i.e.,
\begin{align*}
\imath_{\ell}
& = \operatorname*{arg\,max}_{k \in [n]} \sum_{i = 1}^n (y_i - \bar{y})^2 - \sum_{i = 1}^k (y_{\pi_{\ell}(i)} - S(k,\ell)/k)^2 - \sum_{i = k + 1}^n (y_i - (S(n,\ell) - S(k,\ell))/(n - k))^2 \\
& = \operatorname*{arg\,max}_{k \in [n]} \frac{S(k,\ell)^2}{k} + \frac{(S(n,\ell) - S(k,\ell))^2}{n - k}.
\end{align*}
Fix $\epsilon > 0$. Consider the events
\begin{align*}
A_{\ell}^{\epsilon} & = \bigg\{\max_{k \in [n]} \frac{S(k, \ell)^2}{k} + \frac{(n - S(k, \ell))^2}{n - k} \geq (2 - \epsilon) \log \log(n)\bigg\}, \\
B_{\ell}^{\epsilon} & = \bigg\{\max_{\lfloor n/2 \rfloor \leq k \leq n}\frac{S(k, \ell)^2}{k} + \max_{1 \leq k \leq \lceil n/2 \rceil}\frac{(n - S(k, \ell))^2}{n - k} \leq 2 \epsilon \log \log (n)\bigg\}.
\end{align*}
By \cite[Theorem A.4.1]{csorgo1997limit} $\limsup_{n \rightarrow \infty} \mathbb{P}(A^{\varepsilon}_{\ell}) = \limsup_{n \rightarrow \infty} \mathbb{P}(B^{\varepsilon}_{\ell}) = 1$. Hence for any $\epsilon > 0$,
\begin{align*}
\mathbb{P} \bigg(\sup_{\mathbf{x} \in \mathcal{X}} |\hat{\mu}(\mathbf{x}) - \mu|^2 \geq \frac{(2 - 3 \epsilon) \log \log (n)}{n^b}\bigg) \geq \sum_{\ell = 1}^{p}\mathbb{P}(\mathtt{Imbalance}_{\ell} \cap A_{\ell}^{\epsilon} \cap B_{\ell}^{\epsilon}) \geq \frac{b}{e} + o(1),
\end{align*}
where we have used the fact that $\mathtt{Imbalance}_{\ell}$'s are disjoint for different $\ell$'s and Theorem~\ref{sa-thm:master}. Equation~\eqref{eq:master_rate_constant} then follows.
\begin{center}
\textbf{Part 2: Inconsistency for Points Near the Boundary}
\end{center}
Consider the event
\begin{align*}
\mathtt{Off}_{\ell} & = \{\hat \jmath = \ell, \hat \imath \in [n^a, n^b]\} \\
& = \{\max_k\mathscr{I}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}(k,j), \; \max_{k}\mathscr{I}(k,\ell) > \max_{k \notin [n^a, n - n^b]}\mathscr{I}(k,\ell)\}, \qquad \ell \in [p].
\end{align*}
Since $\pi_{\ell}$ is the uniform permutation, we have
\begin{align*}
\liminf_{n \rightarrow \infty}\mathbb{P}(x_{\ell,\imath_{\ell}} \geq n^{a-1}) \geq \liminf_{n \rightarrow \infty} \mathbb{P}(x_{\ell,\pi_{\ell}(n^a)} \geq n^{a-1}) = 1.
\end{align*}
Together with Theorem~\ref{sa-thm:master},
\begin{align*}
\mathbb{P}(\mathtt{Off}_{\ell}, x_{\ell,\imath_{\ell}} \geq n^{a-1}) \geq \frac{b - a}{2 p e} + o(1).
\end{align*}
Then on the event $\mathtt{Off}_{\ell}$ and $x_{\ell,\imath_{\ell}} \geq n^{a-1}$, for any $\mathbf{x} \in [0,1]^p$ such that $x_{\ell} \leq n^{a-1}$, we have $x_{\ell} \leq x_{\ell,\imath_{\ell}}$, and
\begin{align*}
|\hat\mu(\mathbf{x}) - \mu|^2
& = \frac{S(\imath_{\ell},\ell)^2}{\imath_{\ell}^2} \\
& = \frac{1}{\imath_{\ell}} \bigg(\frac{S(\imath_{\ell},\ell)^2}{\imath_{\ell}} + \frac{(S(n,\ell) - S(\imath_{\ell},\ell))^2}{n - \imath_{\ell}} - \frac{(S(n,\ell) - S(\imath_{\ell},\ell))^2}{n - \imath_{\ell}}\bigg) \\
& \geq \frac{1}{\imath_{\ell}} \bigg(\max_{1 \leq k \leq n}\frac{S(k,\ell)^2}{k} + \frac{(S(n,\ell) - S(k,\ell))^2}{n - k} - \max_{1 \leq k \leq n^b} \frac{(S(n, \ell) - S(k,\ell))^2}{n - k}\bigg).
\end{align*}
By similar arguments as Part 1, we can show
\begin{align*}
\liminf_{n \rightarrow \infty} \inf_{\mathbf{x} \in \mathcal{X}_n}\mathbb{P} \bigg(|\hat{\mu}(\mathbf{x}) - \mu|^2 \geq \frac{(2 + o(1))\log \log(n)}{n^b}\bigg) \geq \frac{b - a}{2 p e},
\end{align*}
which is Equation~\eqref{eq:master_rate_constant2}.
\subsection{Proof of Theorem~\ref{sa-thm: uniform minimax}}
Due to the recursive splitting and Theorem~\ref{sa-thm:master}, the optimal split index $\hat \imath$ at the $K_n$-th split ($K_n \geq 1$) also satisfies
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\big( \hat\imath \leq n^{b}\big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath \big) \geq \frac{b}{2 e}.
\end{align*}
Hence the same argument as Part 1 in the proof of Theorem~\ref{sa-thm:rates} leads to the result.
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency NSS}}
This follows directly from \citet[Theorem 4.3]{klusowski2024large}, choosing $g^{\ast} \equiv \mu$ and $g \equiv \mu$, and changing the sub-Gaussian rate to the sub-exponential rate by choosing $U \asymp \log(n)$ instead of $U \asymp \sqrt{\log(n)}$ in the truncation argument step. The last statement follows from the proof of \citet[Theorem 4.3]{klusowski2024large}.
\subsection{Proof of Theorem~\ref{sa-thm: honest stump}}
Throughout the proof, we abbreviate the honest tree $\hat{\mu}^{\mathtt{HON}}(\mathbf{x})$ by $\check{\mu}(\mathbf{x})$. Recall $(\hat \imath, \hat \jmath)$ denotes the optimal splitting index and coordinate for the decision stump.
We use $(y_i,\mathbf{x}_i^{\top})_{i=1}^M$ to denote $\mathcal{D}_{\mathtt{HON},1}$, which we used to construct the causal tree. Denote by $(\hat \imath, \hat \jmath)$ the splitting index and coordinate at the $K_n$-th step.
Use $(\tilde{y}_i,\tilde{\mathbf{x}}_i^{\top})_{i=1}^N$ to denote $\mathcal{D}_{\mathtt{HON},2}$. By Definition~\ref{sa-defn: cart construction}, $n \lesssim M,N \lesssim n$. Then
\begin{align*}
\sup_{\mathbf{x} \in \mathcal{X}} |\check{\mu}(\mathbf{x}) - \mu| & \geq |\check{\mu}(\mathbf{0}) - \mu| \\
& = \bigg|\frac{\sum_{i = 1}^N (\tilde{y}_i - \mu) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} \bigg|.
\end{align*}
Since $\tilde{y}_i \protect\mathpalette{\protect\independenT}{\perp} \tilde{\mathbf{x}}_i$, condition on $\hat{\imath}, \hat{\jmath}$ and $\mathbf{X} = (\mathbf{x}_1, \cdots, \mathbf{x}_n)$, $\tilde{\mathbf{X}} = (\tilde{\mathbf{x}}_1, \cdots, \tilde{\mathbf{x}}_n)$, we have
\begin{align*}
\frac{\sum_{i = 1}^N (\tilde{y}_i - \mu) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} \stackrel{d}{=} \frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde \imath} (y_i - \mu),
\end{align*}
where
\begin{align*}
\tilde{\imath} = \sum_{i = 1}^N \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath}).
\end{align*}
By Marcinkiewicz–Zygmund inequality, for some positive absolute constant $C$, we have
\begin{align*}
\mathbb{E}\bigg[\Big|\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i - \mu)\Big|\bigg| \hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime} \bigg] & \geq C \mathbb{E} \bigg[\Big|\frac{1}{\tilde{\imath}}\sum_{i = 1}^{\tilde{\imath}} \frac{(y_i - \mu)^2}{\tilde{\imath}}\Big|^{1/2}\bigg| \hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime} \bigg] \\
& \geq C \mathbb{E} \bigg[\frac{1}{\tilde{\imath}}\sum_{i = 1}^{\tilde{\imath}} \Big|\frac{(y_i - \mu)^2}{\tilde{\imath}}\Big|^{1/2}\bigg| \hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime} \bigg] \\
& \geq \frac{C\mathbb{E}[|y_i - \mu|]}{\tilde{\imath}^{1/2}},
\end{align*}
where in the second to last line, we have used Jensen's inequality, and in the last line we have used $\tilde{\imath}$ is measurable with resepect to the $\sigma$-algebra generated by $\hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}$. Then by Paley–Zygmund inequality, for any $\theta \in (0,1)$,
\begin{align*}
\mathbb{P} \bigg(\Big|\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i & - \mu)\Big| \geq \theta \frac{C\mathbb{E}[|y_i - \mu|]}{\tilde{\imath}^{1/2}} \bigg|\hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}\bigg) \\
& \geq \mathbb{P} \bigg(\Big|\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i - \mu)\Big| \geq \theta \mathbb{E} \Big[\Big|\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i - \mu)\Big|\hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}\Big]\bigg|\hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}\bigg) \\
& \geq (1 - \theta)^2 \frac{\mathbb{E} \Big[|\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i - \mu)|\Big| \hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}\Big]^2}{\mathbb{E} \Big[\Big(\frac{1}{\tilde{\imath}} \sum_{i = 1}^{\tilde{\imath}} (y_i - \mu)\Big)^2\Big|\hat \imath, \hat \jmath, \mathbf{X}, \mathbf{X}^{\prime}\Big]} \\
& \geq C (1 - \theta)^2 \frac{\mathbb{E}[|y_i - \mu|^2]}{\mathbb{V}[y_i]}.
\end{align*}
Now we want to obtain a high probability upper bound on $\tilde \imath$ given $\imath$. Let $F$ be the cummulative distribution function of $\mathbf{x}_i$. Suppose $1 \leq k \leq N/2$. Then $F(\mathbf{x}_{(k)}) \sim \mathsf{Beta}(k, M - k + 1)$. By a Bernstein bound for Beta variables \cite[Theorem 1]{skorski_2023}, we have for all $\epsilon > 0$,
\begin{align*}
\mathbb{P}(F(\mathbf{x}_{(k)}) > k/M + \epsilon) \leq \exp \Big( - \frac{\epsilon^2}{2 v + \frac{c \epsilon}{3}}\Big),
\end{align*}
where for large enough $n$,
\begin{align*}
v & = \frac{k (M - k + 1)}{(M + 1)^2 (M + 2)} \leq 2\frac{k}{M^2}, \\
c & = \frac{2 (M - 2k + 1)}{M (M + 2)} \leq \frac{2}{M}.
\end{align*}
Hence with probability at least $1 - M^{-1}$,
\begin{align*}
F(\mathbf{x}_{(k)}) \leq k/M + 2\frac{\sqrt{\log (M) k}}{M} + 3 \frac{\log (M)}{M}.
\end{align*}
Condition on $\mathbf{X}$, $\mathds{1}(\tilde{\mathbf{x}}_i \leq \mathbf{x}_{(k)})$'s are i.i.d $\mathsf{Bernoulli}(F(\mathbf{x}_{(k)}))$. Hence condition on $\mathbf{X}$ and $\hat{\imath}$, with probability at least $1 - N^{-1}$,
\begin{align*}
\tilde{\imath}/N = n^{-1}\sum_{i = 1}^N \mathds{1}(\tilde{\mathbf{x}}_i \leq \mathbf{x}_{(\hat{\imath})}) \leq F(\mathbf{x}_{(\hat{\imath})}) + 2 \sqrt{\frac{\log(N) F(\mathbf{x}_{(\hat{\imath})})}{N}}.
\end{align*}
Hence condition on the event $\hat \imath \leq M^b$, we have with probability at least $1 - 2 N^{-1}$,
\begin{align*}
\tilde{\imath}/n \leq 4 M^{b-1} \leq C n^{b-1},
\end{align*}
where $C$ is some constant only depending on $\liminf_{n \to \infty}|D_{\mathtt{HON}, 1}|/|\mathcal{D}_{\mathtt{HON},2}|$ and $\limsup_{n \to \infty}|D_{\mathtt{HON}, 1}|/|\mathcal{D}_{\mathtt{HON},2}|$.
Due to the iterative partitioning, the conclusion for Theorem~\ref{sa-thm:master} holds not only for decision stump, but also for the splitting index at arbitrary depth $K_n$, that is, for any $b \in (0,1)$, we have
\begin{align*}
\liminf_{M \rightarrow \infty}\mathbb{P}(\hat \imath \leq M^b) \geq \frac{b}{2e}.
\end{align*}
Hence we have
\begin{align*}
\mathbb{P} \bigg(|\hat{\mu}(\mathbf{0}) - \mu| \geq \theta \frac{C\mathbb{E}[|y_i - \mu|]}{C n^{b/2}} \bigg)
& \geq \sum_{k \leq M^b} \mathbb{P}\bigg(|\hat{\mu}(\mathbf{0}) - \mu| \geq \theta \frac{C\mathbb{E}[|y_i - \mu|]}{C n^{b/2}} \bigg| \hat{\imath} = k \bigg) \mathbb{P}(\hat{\imath} = k) \\
& \geq \sum_{k \leq M^b} \mathbb{P}\bigg(|\hat{\mu}(\mathbf{0}) - \mu| \geq \theta \frac{C\mathbb{E}[|y_i - \mu|]}{{\tilde \imath}^{1/2}} \bigg| \hat{\imath} = k \bigg) \mathbb{P}(\hat{\imath} = k) - 2 n^{-1}\\
& \geq C (1 - \theta)^2 \frac{\mathbb{E}[|y_i - \mu|^2]}{\mathbb{V}[y_i]} \frac{b}{2 e} - 2 n^{-1}.
\end{align*}
This proves the conclusion.
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency honest}}
For notational simplicity, we use $\mathsf{T}$ to denote the data-driven decision tree. We will follow the proof strategy from \citet[Theorem 4.3]{klusowski2024large} condition on $\mathcal{D}_{\mathsf{T}}$. Denote by $\mathscr{G}_0$ the class of constant functions. Decompose $\lVert \hat{\mu}(\mathsf{T}) - \mu \rVert^2 = E_1 + E_2$, where
\begin{align*}
E_1 = \lVert \hat{\mu}(\mathsf{T}) - \mu \rVert^2 - 2 (\lVert y - \hat{\mu}(\mathsf{T})\rVert^2_{\mathcal{D}_{\mu}} - \lVert y - \mu\rVert^2_{\mathcal{D}_{\mu}}) - \alpha - \beta,
\end{align*}
and
\begin{align*}
E_2 = 2 (\lVert y - \hat{\mu}(\mathsf{T})\rVert^2_{\mathcal{D}_{\mu}} - \lVert y - \mu\rVert^2_{\mathcal{D}_{\mu}}) + \alpha + \beta.
\end{align*}
Denote the partition for $\mathsf{T}$ by $\mathcal{P}$. Since $\mathcal{P}$ is independent to $\mathcal{D}_{\mu}$, the bound (E.27) from \citet{klusowski2024large} does not apply automatically. Instead, we consider $\mathscr{G}_0$ as the reference class. Given the partitions of $\mathsf{T}$, the values of leaf nodes are obtained by least-square projection using $\mathcal{D}_{\mu}$. This immediately implies
\begin{align*}
\lVert y - \hat{\mu}(\mathsf{T})\rVert^2_{\mathcal{D}_{\mu}} \leq \lVert y - \bar{y}\rVert^2_{\mathcal{D}_{\mu}} \leq
\lVert y - g \rVert^2_{\mathcal{D}_{\mu}},
\end{align*}
for any constant function $g \in \mathscr{G}_0$. Hence for all $g \in \mathscr{G}_0$,
\begin{align*}
\mathbb{E}_{\mathcal{D}_{\mu}}[E_2|\mathcal{D}_{\mathsf{T}}] & \leq 2 \mathbb{E}_{\mathcal{D}_{\mu}}[\lVert y - g\rVert^2_{\mathcal{D}_{\mu}} - \lVert y - \mu\rVert^2_{\mathcal{D}_{\mu}}|\mathcal{D}_{\mathsf{T}}] + \alpha + \beta \\
& = 2 \lVert g - \mu \rVert^2 + \alpha + \beta.
\end{align*}
For the term $E_1$, we first assume $|y_i| \leq U$. Observe that condition on $\mathcal{D}_{\mathsf{T}}$, $\hat{\mu}(\mathsf{T})$ is still a member of the class $\mathscr{G}_{n_{\mathsf{T}}}[\mathcal{P}]$, which is the collection of all piecewise constant functions (bounded by $U$) on the partition $\mathcal{P}$. Since for any $\varepsilon \in (0,1)$,
\begin{align*}
N(\varepsilon U, \mathscr{G}_{n_{\mathsf{T}}}[\mathcal{P}], \lVert \cdot \rVert_{P_{X^{n_{\mu}}}})
\leq N(\varepsilon U, \mathscr{G}_{n_{\mathsf{T}}}[\mathcal{P}], \lVert \cdot \rVert_{\infty})
\leq \bigg(\frac{2}{\varepsilon}\bigg)^{2^K},
\end{align*}
we can still use \citet[Theorem 11.4]{Gyorfi-etal_2002_book} and the same argument from Equation (B.30) to (B.33) in \citet{klusowski2024large} to get
\begin{align*}
\mathbb{P}_{\mathcal{D}_{\mu}}(E_1 \geq 0 \;|\;\mathcal{D}_{\mathsf{T}}) \leq 14 \bigg(\frac{2 U^2}{\beta}\bigg)^{2^{K}} \exp \bigg(- \frac{\alpha n_{\mathsf{T}}}{2568 U^4} \bigg).
\end{align*}
The result then follows choosing $\alpha \asymp \frac{U^4 2^K \log(n)}{n}$ and $\beta \asymp \frac{U^2}{n}$, and truncation argument over the sub-exponential $\varepsilon_i$'s.
\subsection{Proof of Theorem~\ref{thm:main}}
In this section, we prove Theorem \ref{thm:main}. First, we define some notation related to the tree construction which will be used in the proofs. Let $\tilde n_k$ be the number of observations in the node containing $x = 0$ at depth $k$, $\tilde\imath_{k} $ be the CART split index of this node, and $\tilde \jmath_{k}$ be the CART split coordinate of this node, with $\tilde n_0 = n$ and $\tilde\imath_0 = \hat\imath$ (recall that $ \hat\imath $ is the split index for the decision stump \eqref{eq:stump}). Then, the left-most cell at the $k$-th level can be expressed as $ \mathbf{t} \cap [0, x_{ \pi_{\tilde{\jmath},\tilde{\jmath}}(\tilde\imath_{k-1})}] $ and $ \tilde n_k = \tilde \imath_{k-1}$.
\begin{lemma} \label{lmm:recursion}
There exist $ \delta \in (0, 1) $, $c > 1$, and a positive integer $M$ such that for any depth $k\geq 1$ and $m\geq M$, we have
$
\mathbb{P}(\tilde n_k \leq m) \geq (1-\delta)\cdot \mathbb{P}(\tilde n_{k-1}\leq m) + \delta \cdot \mathbb{P}(\tilde n_{k-1}\leq m^c).
$
\end{lemma}
\begin{proof}
Observe that if $m$ is a positive integer, then $\tilde\imath_{k-1} \mid \tilde n_{k-1} = m$ has the same distribution as $\tilde\imath_0 \mid \tilde n_0 = m$, because of the honest tree construction and Assumption \ref{sa-ass:DGP}.
Therefore, we can apply \eqref{eq:split_range} to obtain
\begin{equation} \label{eq:conditional_split}
\mathbb{P}\big( m^a \leq \tilde\imath_{k-1} \leq
m^b \mid \tilde n_{k-1} = m\big) \geq \delta > 0,
\end{equation}
for some $ \delta > 0 $ and sufficiently large $ m $. Hence, by \eqref{eq:conditional_split}, we have for $ m $ sufficiently large,
\begin{equation} \label{eq:cond}
\begin{aligned}
& \mathbb{P}\big(\tilde n_k \leq m \mid m < \tilde n_{k-1} \leq m^{1/b} \big) \\ &
\quad \geq \min_{m < i \leq m^{1/b} } \mathbb{P}\big(i^a \leq \tilde\imath_{k-1}\leq i^b \mid \tilde n_{k-1} = i\big)\mathbb{P}\big(\tilde n_k \leq m \mid i^a \leq \tilde\imath_{k-1} \leq i^b \big) \\
& \quad \geq \delta \min_{m < i \leq m^{1/b} } \mathbb{P}\big(\tilde n_k \leq m \mid i^a \leq \tilde\imath_{k-1} \leq i^b \big) \\
& \quad \geq \delta\min_{m^a < i \leq m } \mathbb{P} \big(\tilde n_k \leq \tilde\imath_{k-1} \mid \tilde\imath_{k-1} = i \big) \\
& \quad = \delta.
\end{aligned}
\end{equation}
Now, taking $ c = 1/b $, note that
\eqref{eq:cond} implies Lemma \ref{lmm:recursion} since, for $ m $ sufficiently large, we have
\begin{align}
\mathbb{P}(\tilde n_k \leq m)
& = \mathbb{P}(\tilde n_k \leq m, \; \tilde n_{k-1} > m^c) + \mathbb{P}(\tilde n_k \leq m, \; \tilde n_{k-1} \leq m^c) \\
& \geq \mathbb{P}(\tilde n_k \leq m, \; \tilde n_{k-1} \leq m^c) \\
& = \mathbb{P}(\tilde n_k \leq m, \; \tilde n_{k-1} \leq m) + \mathbb{P}(\tilde n_k \leq m, \; m < \tilde n_{k-1} \leq m^c) \\
& \geq \mathbb{P}(\tilde n_{k-1}\leq m)+ \delta\cdot\mathbb{P}(m < \tilde n_{k-1}\leq m^c) \\
& = (1-\delta)\cdot \mathbb{P}(\tilde n_{k-1}\leq m)+ \delta\cdot\mathbb{P}(\tilde n_{k-1}\leq m^c). \qedhere
\end{align}
\end{proof}
Next, we use Lemma \ref{lmm:recursion} to finish the proof of Theorem \ref{thm:main}. The main idea is to establish that the terminal nodes in a shallow tree will be small with constant probability.
\begin{proof}[Proof of Theorem \ref{thm:main}]
For notational simplicity, we denote $\hat \mu^{\mathtt{X}}(\mathbf{x};K)$ by $\tilde{\mu}(T_K)(\mathbf{x})$.
Define $n_\ell = N^{(1/c)^{\ell}}$, where $N = n/K_n$.
We will show by induction that for any $k \geq 0$ and $\ell \geq 1$ such that $n_\ell \geq M$,
\begin{equation} \label{eq:induction}
\mathbb{P}(\tilde{n}_k \leq n_{\ell}) \geq
\sum_{k'=\ell}^{k} \binom{k'-1}{\ell-1} (1-\delta)^{k'-\ell} \delta^\ell..
\end{equation}
The base case of $k = 0$ is trivial since $\tilde n_0 = N$.
Now, assume that for some fixed $k \geq 1$ and any $\ell' \geq 1$ such that $n_{\ell'} \geq M$, we have
\begin{equation} \label{eq:hypothesis}
\mathbb{P}(\tilde n_{k-1}\leq n_{\ell'}) \geq
\sum_{k'=\ell'}^{k-1} \binom{k'-1}{\ell'-1} (1-\delta)^{k'-\ell'} \delta^{\ell'}.
\end{equation}
If $\ell \geq 2$, then substituting our induction hypothesis \eqref{eq:hypothesis} with $\ell' = \ell$ and $\ell'= \ell-1$ into Lemma \ref{lmm:recursion}, we get that
\begin{align}
\mathbb{P}(\tilde n_k \leq n_\ell)
& \geq (1-\delta) \sum_{k'=\ell}^{k-1} \binom{k'-1}{\ell-1} (1-\delta)^{k'-\ell} \delta^\ell + \delta \sum_{k'=\ell-1}^{k-1} \binom{k'-1}{\ell-2} (1-\delta)^{k'-\ell+1}\delta^{\ell-1} \\
& =\sum_{k'=\ell}^k \binom{k'-1}{\ell-1} (1-\delta)^{k'-\ell} \delta^\ell,
\end{align}
where we used Pascal's identity.
This completes the inductive proof of \eqref{eq:induction}.
Let $X \sim \text{NB}(L, \delta)$, i.e., the number of independent trials, each occurring with probability $\delta$, until $L$ successes. Choose
$$
L = \lceil \log_c\log_c(N) - \log_c\log_c(M)-1\rceil \asymp \log\log(N), \quad n_L = N^{(1/c)^L} \in [M,M^c].
$$
By \eqref{eq:induction} and Markov's inequality applied to the tail probability of $ X $, we have that
\begin{equation}
\begin{aligned} \label{eq:terminal}
\mathbb{P}(\tilde n_K \leq n_L) & \geq
\sum_{k'=L}^K \binom{k'-1}{L-1} (1-\delta)^{k'-L}\delta^L \\ & = 1 - \mathbb{P}(X \geq K + 1) \\ & \geq 1- \frac{\mathbb{E}[X]}{K+1} \\ & = 1 - \frac{L}{\delta(K+1)} \\ & \geq \frac{1}{2},
\end{aligned}
\end{equation}
as long as $ K \geq 2L/\delta \gtrsim \log\log(N)$. By the Paley-Zygmund inequality \citep{petrov2007lower} and the fact that $ \text{Var}(\tilde \mu(T_{K})(0)) = \mathbb{E}[1/\tilde n_K] \leq 1 $,
we have
\begin{equation} \label{eq:paley}
\mathbb{P}\bigg(|\tilde \mu(T_{K})(0)| > \frac{\mathbb{E}[|\tilde \mu(T_{K})(0)|]}{2} \bigg) \geq \frac{(\mathbb{E}[|\tilde \mu(T_{K})(0)|])^2}{4\text{Var}(\tilde \mu(T_{K})(0))} \geq \frac{(\mathbb{E}[|\tilde \mu(T_{K})(0)|])^2}{4} .
\end{equation}
By the honest construction of the tree and \eqref{eq:terminal}, we have the lower bound
\begin{equation}
\begin{aligned} \label{eq:expect_paley}
\mathbb{E}[|\tilde \mu(T_{K})(0)|] & = \sum_{k=1}^n \mathbb{E}\bigg[\bigg|\frac{1}{k}\sum_{i=1}^k \tilde y_i \bigg|\bigg]\mathbb{P}( \tilde n_K = k) \\
& \geq \min_{ k = 1, 2, \dots, \lceil n_L \rceil }\mathbb{E}\bigg[\bigg|\frac{1}{k}\sum_{i=1}^k \tilde y_i \bigg|\bigg]\mathbb{P}( \tilde n_K \leq \lceil n_L \rceil) \\ & \geq \frac{1}{2}\min_{ k = 1, 2, \dots, \lceil n_L \rceil}\mathbb{E}\bigg[\bigg|\frac{1}{k}\sum_{i=1}^k \tilde y_i \bigg|\bigg].
\end{aligned}
\end{equation}
Since a sum of independent random variables is almost surely constant if and only if the individual random variables are almost surely constant, it follows that the last expression in \eqref{eq:expect_paley} is bounded away from zero. Returning to \eqref{eq:paley} completes the proof.
\end{proof}
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency X}}
For simplicity, denote $\hat{\mu}^{\mathtt{X}}(\mathbf{x})$ by $\tilde \mu(T_K)(\mathbf{x})$, and $N = n/(K+1)$ denotes the sample size for each folds in the $\mathtt{X}$ sample splitting scheme.
Let $\mathsf{t}_1, \mathsf{t}_2, \cdots, \mathsf{t}_{2^K}$ denote the $2^K$ leaf nodes in the decision tree, (if a node cannot be further refined, we duplicate the split indices and values at the next level). And let $N_1, N_2, \cdots, N_{2^K}$ and $m_1, m_2, \cdots, m_{2^K}$ denote the number of observations and the Lebesgue measure of the $2^K$ leaf nodes, respectively. Note that $\Vec{N} = (N_1, \cdots, N_{2^{K}})$ are independent of the $\tilde y_i $ data by the honest condition and the $x_i $ data per Assumption \ref{sa-ass:DGP}.
\textit{Claim:} Condition on $\Vec{N}$, $m_k \sim \mathsf{Beta}(N_k, N - N_k + 1)$
Thus, the IMSE can be bounded as follows:
\begin{align*}
\mathbb{E}\Bigg[\int_{\mathcal{X}}(\tilde \mu(T_K)(x))^2 \mathbb{P}_x(dx)\Bigg] & =
\sum_{k=1}^{2^K}\mathbb{E}\Bigg[m_k\Bigg(\frac{\mathds{1}(N_k > 0)}{N_k}\sum_{i=1}^n \tilde y_i\mathds{1}(\mathbf{x}_i \in \mathsf{t}_k)\Bigg)^2\Bigg]
\\ & =
\sum_{k=1}^{2^K}\mathbb{E}\Bigg[\frac{m_k}{N_k}\mathds{1}(N_k > 0)\Bigg] \sigma^2 \\
& \leq \sum_{k=1}^{2^K}\mathbb{E}\Bigg[\frac{1}{N+1}\Bigg] \sigma^2 \\
& \leq \frac{2^{K+1}}{N+1} \sigma^2.
\qedhere
\end{align*}
\textit{Proof of Claim:} We show by induction. \underline{Base Case: K = 1.} For decision stumps, for some coordinate $j \in [p]$, we have $m_1 = x_{(N_1)}$, and $m_2 = x_{(N_1 + N_2)} - x_{(N_1)} = 1 - x_{(N_1)}$. By Assumption~\ref{sa-ass:DGP}, the order statistics $x_{j,(i)}$ is independent to $\Vec{N}$. Hence $m_k \sim \mathsf{Beta}(N_k, N - N_k + 1)$, $k = 1,2$.
\underline{Induction Step: $K \geq 2$.} Let $\mathsf{t}_l^{\text{prev}}$ be a $(K - 1)$-th level node, we annonate all relevant depth $K-1$ information with superscript $\text{prev}$. We already know condition on $N_1^{\text{prev}}, \cdots, N_{2^{K-1}}^{\text{prev}}$, $m_l^{\text{prev}} \sim \mathsf{Beta}(N_l^{\text{prev}}, N - N_l^{\text{prev}} + 1)$. Suppose $\mathsf{t}_l^{\text{prev}}$ is divided into $\mathsf{t}_{2l}, \mathsf{t}_{2l+1}$ with Lebesgue measure and number of observations given by $m_{2l}, m_{2l+1}$ and $N_{2l}$, $N_{2l+1}$, respectively, and the split is based on coordinate $j \in [p]$. By Assumption \ref{sa-ass:DGP}, condition on $\mathbf{x}_i \in \mathsf{t}_l^{\text{prev}}$, $\mathbf{x}_i \sim \mathsf{Uniform}(\mathsf{t}_l^{\text{prev}})$. Hence condition on $N_l^{\text{prev}}$, $N_{2l}$ and $m_l^{\text{prev}}$, we have $m_{2l}/m_{l}^{\text{prev}} \sim \mathsf{Beta}(N_l^{\text{prev}}, N - N_l^{\text{prev}} + 1)$. Hence condition on $\vec{N} = (N_1, \cdots, N_{2^K})$, we have $m_k \sim \mathsf{Beta}(N_k, N - N_k + 1)$, $1 \leq k \leq 2^K$. Induction then concludes the proof.
\subsection{Proof of Corollary~\ref{coro: inconsistency ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm:master}.
\subsection{Proof of Corollary~\ref{coro: rates ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm:rates}.
\subsection{Proof of Corollary~\ref{sa-coro: uniform minimax ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm: uniform minimax}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency NSS ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm: L2 consistency NSS}.
\subsection{Proof of Corollary~\ref{sa-coro: honest output ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm: honest stump}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency honest ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm: L2 consistency honest}.
\subsection{Proof of Corollary~\ref{sa-coro: main ipw}}
This is an immediate corollary from Theorem~\ref{thm:main}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency ipw}}
This is an immediate corollary from Theorem~\ref{sa-thm: L2 consistency X}.
\subsection{Proof of Lemma~\ref{lem: approximation -- balanced region}}
Since the number of coordinate $p$ is fixed, we can use a union bound over the approximation error for the $p$ coordinates. Hence w.lo.g. we can assume $p = 1$ and drop the second index on the coordinate $\ell$ from $\mathscr{I}^{\mathtt{DIM}}(k,\ell)$ and $\bar \mathscr{I}^{\mathtt{IPW}}(k,\ell)$ everywhere. And throughout, we assume the data is already sorted so that
\begin{align*}
x_1 \leq x_2 \leq \cdots \leq x_n.
\end{align*}
Expand the square, we have for any $k = 1,2,\cdots,n$,
\begin{align}\label{eq:expand square}
\mathscr{I}^{\mathtt{DIM}}(k) - \bar \mathscr{I}^{\mathtt{IPW}}(k) & = \frac{k (n - k)}{n} \underbrace{\Big(\hat\tau^{\mathtt{DIM}}_{\mathrm{t}_L}(k)-\hat\tau^{\mathtt{DIM}}_{\mathrm{t}_R}(k) + \bar\tau^{\mathtt{IPW}}_{\mathrm{t}_L}(k)-\bar\tau^{\mathtt{IPW}}_{\mathrm{t}_R}(k)\Big)}_{=:R_1(k)} \underbrace{\Big(\hat\tau^{\mathtt{DIM}}_{\mathrm{t}_L}(k)-\hat\tau^{\mathtt{DIM}}_{\mathrm{t}_R}(k) - \bar\tau^{\mathtt{IPW}}_{\mathrm{t}_L}(k)+\bar\tau^{\mathtt{IPW}}_{\mathrm{t}_R}(k)\Big)}_{=:R_2(k)}.
\end{align}
We focus on the case where $1 \leq k \leq \frac{n}{2}$, the other case where $\frac{n}{2} < k \leq n$ follow from symmetry. Consider the term $R_2(k)$. First, consider the term corresponding to $i$ from $1$ to $k$. The other term corresponding to $i$ from $k + 1$ to $n$ can be handled similarly. Breaking down $y_i(1) = \mu_1(x_i) + \varepsilon_i(1)$ and $y_i(0) = \mu_0(x_i) + \varepsilon_i(0)$, we have
\begin{align}\label{eq: decomp}
\nonumber |R_2(k)| & = \bigg|\frac{\sum_{i = 1}^k d_i y_i(1)}{\sum_{i = 1}^k d_i} - \frac{1}{k}\sum_{i = 1}^k \frac{d_i}{\xi} \varepsilon_i(1) - \frac{\sum_{i = 1}^k (1 - d_i) y_i(0)}{\sum_{i = 1}^k (1 - d_i)} + \frac{1}{k}\sum_{i = 1}^k \frac{1 - d_i}{1 - \xi} \varepsilon_i(0) + \text{counterpart for $t_R$}\bigg| \\
\nonumber & \leq \bigg|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i}\bigg| \cdot \bigg|\frac{1}{k}\sum_{i = 1}^k (\frac{d_i}{\xi} - 1) \bigg| + \bigg|\frac{\sum_{i = 1}^k (1 - d_i) \varepsilon_i(0)}{\sum_{i = 1}^k (1-d_i)}\bigg| \cdot \bigg|\frac{1}{k}\sum_{i = 1}^k (\frac{1 -d_i}{1 -\xi} - 1) \bigg| \\
\nonumber & \quad +\bigg|\frac{\sum_{i = k+1}^n d_i \varepsilon_i(1)}{\sum_{i = k+1}^n d_i}\bigg| \cdot \bigg|\frac{1}{n - k}\sum_{i = k+1}^n (\frac{d_i}{\xi} - 1) \bigg| + \bigg|\frac{\sum_{i = k+1}^n (1 - d_i) \varepsilon_i(0)}{\sum_{i = k+1}^n (1-d_i)}\bigg| \cdot \bigg|\frac{1}{n - k}\sum_{i = k+1}^n (\frac{1 -d_i}{1 -\xi} - 1) \bigg| \\
& \quad + \bigg|\frac{\sum_{i =1}^k d_i \mu_1(x_i)}{\sum_{i =1}^k d_i} - \frac{\sum_{i = 1}^k (1-d_i) \mu_0(x_i)}{\sum_{i = 1}^k (1 -d_i) } - \frac{\sum_{i =k+1}^n d_i \mu_1(x_i)}{\sum_{i = k+1}^n d_i} + \frac{\sum_{i = k+1}^n (1-d_i) \mu_0(x_i)}{\sum_{i = k+1}^n (1 -d_i) }\bigg|.
\end{align}
Notice that Assumption~\ref{sa-assump: dgp-causal} (ii) implies that the last term is zero. Since $x_i \protect\mathpalette{\protect\independenT}{\perp} d_i$, even though the data is ordered according to $x_i$, $\{d_i/\xi - 1: 1 \leq i \leq n\}$ are i.i.d mean-zero with bounded second moment. By Theorem A.4.1 in \cite{csorgo1997limit},
\begin{align*}
\max_{r_n \leq k < n - r_n} \sqrt{k} \cdot \bigg|\frac{1}{k}\sum_{i = 1}^k (\frac{d_i}{\xi} - 1)\bigg|
= O_{\mathbb{P}} (\sqrt{\log \log(n)}).
\end{align*}
Take $b_i = \sum_{1 \leq \ell \leq i} d_\ell$. By Equation (8) from \cite{shorack1976inequalities}, for any $\lambda > 0$,
\begin{align*}
\mathbb{P} \bigg(\max_{r_n \leq k \leq n - r_n} \Big|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\Big| \geq \lambda \bigg| (d_i)_{1 \leq i \leq n}\bigg) & \leq 16 \sum_{r_n \leq i \leq n - r_n} \frac{d_i\mathbb{V}[\varepsilon_i(1)]}{b_i^2} \lambda^{-2} \\
& \leq 16 \sum_{i \geq b_{r_n}}\frac{1}{i^2} \lambda^{-2} \mathbb{V}[\varepsilon_i(1)] \\
& \leq \frac{8}{3}\pi^2 \lambda^{-2} \mathbb{V}[\varepsilon_i(1)] \frac{1}{b_{r_n}},
\end{align*}
The assumption that $\liminf_{n \rightarrow \infty} \rho_n \log \log (n) = \infty$ implies $\liminf_{n \rightarrow \infty} r_n = \infty$. Hence
\begin{align*}
(b_{r_n})^{-1} = r_n^{-1} \Big(\xi + \frac{1}{r_n}\sum_{i = 1}^{r_n}(d_i - \xi)\Big)^{-1} = O_\mathbb{P}(r_n^{-1}).
\end{align*}
Hence uncondition on $(d_i)_{1 \leq i \leq n}$, and we have
\begin{align}\label{eq: delayed partial sum}
\max_{r_n \leq k \leq n - r_n} \Big|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\Big| = O_\mathbb{P}(r_n^{-1/2}).
\end{align}
Hence
\begin{align*}
\max_{r_n \leq k < n - r_n} \sqrt{k} \cdot \bigg|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i}\bigg| \cdot \bigg|\frac{1}{k}\sum_{i = 1}^k (\frac{d_i}{\xi} - 1) \bigg| = O_{\mathbb{P}} \bigg(\sqrt{\frac{\log \log (n)}{r_n}}\bigg).
\end{align*}
By similar arguments, we can show the same bound holds for other terms in the first two lines of Equation~\eqref{eq: decomp}. Hence
\begin{align*}
\max_{r_n \leq k < n - r_n} \sqrt{k}|R_2(k)| = O_\mathbb{P} \bigg( \sqrt{\frac{\log \log (n)}{r_n}}\bigg).
\end{align*}
Under the assumption that $\mu_0 \equiv c_0$ and $\mu_1 \equiv c_1$, we have
\begin{align*}
\nonumber R_1(k) & = \bigg|\frac{\sum_{i = 1}^k d_i y_i}{\sum_{i = 1}^k d_i} + \frac{1}{k}\sum_{i = 1}^k \frac{d_i}{\xi} \varepsilon_i(1) - \frac{\sum_{i = 1}^k (1 - d_i) y_i}{\sum_{i = 1}^k (1 - d_i)} - \frac{1}{k}\sum_{i = 1}^k \frac{1 - d_i}{1 - \xi} \varepsilon_i(0) + \text{counterpart for $t_R$}\bigg| \\
& = \bigg|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i} + \frac{1}{k}\sum_{i = 1}^k \frac{d_i}{\xi} \varepsilon_i(1) - \frac{\sum_{i = 1}^k (1 - d_i) \varepsilon_i(0)}{\sum_{i = 1}^k (1 - d_i)} - \frac{1}{k}\sum_{i = 1}^k \frac{1 - d_i}{1 - \xi} \varepsilon_i(0) + \text{counterpart for $t_R$}\bigg|.
\end{align*}
By Equation~\eqref{eq: delayed partial sum} and Theorem A.4.1 in \cite{csorgo1997limit} for the terms $k^{-1}\sum_{i = 1}^k \xi^{-1} d_i \varepsilon_i(1)$, $k^{-1} \sum_{i = 1}^k (1 - \xi)^{-1} (1 - d_i) \varepsilon_i(0)$ and the counterparts for $t_R$, we have
\begin{align*}
\max_{r_n \leq k < n - r_n}\sqrt{k}|R_1(k)| = O_\mathbb{P} \bigg(\sqrt{\log \log (n)}\bigg).
\end{align*}
Putting together the parts for $R_1$ and $R_2$, we have
\begin{align*}
\max_{r_n \leq k < n - r_n}|\mathscr{I}^{\mathtt{DIM}}(k) - \bar \mathscr{I}^{\mathtt{IPW}}(k)| = O_\mathbb{P} \bigg( \frac{\log \log (n)}{r_n^{1/2}}\bigg).
\end{align*}
\subsection{Proof of Lemma~\ref{lem: approximation -- imbalanced region}}
Since the number of coordinates $p$ is fixed, we can use a union bound over the approximation error for the $p$ coordinates. Hence w.lo.g. we can assume $p = 1$ and drop the second index on the coordinate $\ell$ from $\mathscr{I}^{\mathtt{DIM}}(k,\ell)$ and $\bar \mathscr{I}^{\mathtt{IPW}}(k,\ell)$ everywhere. And throughout, we assume the data is already sorted so that
\begin{align*}
x_1 \leq x_2 \leq \cdots \leq x_n.
\end{align*}
For $1 \leq k \leq s_n$ and $n - s_n \leq k \leq n$, Equations~\eqref{eq:expand square} and \eqref{eq: decomp} still hold. W.l.o.g assume $1 \leq k \leq s_n$. First, we upper bound the IPW terms. Definition of $s_n$ and Equation (A.4.3) in \cite{csorgo1997limit} imply
\begin{align}\label{eq: imbalanced ipw partial sum}
\max_{1 \leq k \leq s_n} \bigg|\frac{1}{\sqrt{k}}\sum_{i = 1}^k \frac{d_i}{\xi}\varepsilon_i(1)\bigg| + \bigg| \frac{1}{\sqrt{k}} \sum_{i =1}^k \frac{1 - d_i}{1 - \xi} \varepsilon_i(0)\bigg| = O_\mathbb{P} (u_n),
\end{align}
with $u_n = \sqrt{\rho_n \log \log(n)}$. Also Equation (A.4.2) in \cite{csorgo1997limit} imply
\begin{align}\label{eq: imbalanced ipw partial sum large}
\max_{1 \leq k \leq s_n} \sqrt{k} \cdot \bigg|\frac{1}{n-k}\sum_{i = k+1}^n \frac{d_i}{\xi}\varepsilon_i(1)\bigg| + \sqrt{k} \cdot \bigg| \frac{1}{n-k} \sum_{i = k+1}^n \frac{1 - d_i}{1 - \xi} \varepsilon_i(0)\bigg| = O_\mathbb{P}(v_n),
\end{align}
where $v_n = \sqrt{\frac{s_n}{n - s_n} \log \log (n)}$. Again Equation (A.4.3) from \cite{csorgo1997limit} imply that
\begin{align*}
\max_{1 \leq k \leq s_n} \bigg|\frac{1}{\sqrt{k}} \sum_{i = 1}^k \Big(\frac{d_i}{\xi} - 1 \Big) \bigg| = O_\mathbb{P}(u_n).
\end{align*}
Take $b_i = \sum_{1 \leq \ell \leq i} d_\ell$. By Equation (8) from \cite{shorack1976inequalities}, for any $\lambda > 0$,
\begin{align*}
\mathbb{P} \bigg(\max_{1 \leq k \leq s_n} \Big|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\Big| \geq \lambda \bigg| (d_i)_{1 \leq i \leq n}\bigg) & \leq 16 \sum_{1 \leq i \leq s_n} \frac{d_i\mathbb{V}[\varepsilon_i(1)]}{b_i^2} \lambda^{-2} \\
& \leq 16 \sum_{1 \leq i \leq s_n}\frac{1}{i^2} \lambda^{-2} \mathbb{V}[\varepsilon_i(1)] \\
& \leq \frac{8}{3}\pi^2 \lambda^{-2} \mathbb{V}[\varepsilon_i(1)],
\end{align*}
Hence uncondition on $(d_i)_{1 \leq i \leq n}$, and we have
\begin{align*}
\max_{1 \leq k \leq s_n} \Big|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\Big| = O_\mathbb{P}(1).
\end{align*}
It follows that
\begin{align}\label{eq: bound}
\max_{1 \leq k \leq s_n} \sqrt{k} \cdot\bigg|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i} - \frac{1}{k}\sum_{i = 1}^k \frac{d_i}{\xi} \varepsilon_i(1)\bigg|
= \max_{1 \leq k \leq s_n} \bigg|\frac{1}{\sqrt{k}} \sum_{i = 1}^k \Big(\frac{d_i}{\xi} - 1 \Big) \cdot \frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i} \bigg| = O_\mathbb{P} (u_n).
\end{align}
Putting together the above equation with Equation~\eqref{eq: imbalanced ipw partial sum} and using a similar argument for the control group,
\begin{align}\label{eq: imbalanced reg partial sum}
\max_{1 \leq k \leq s_n} \sqrt{k} \cdot \bigg|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i = 1}^k d_i}\bigg| + \sqrt{k} \cdot \bigg|\frac{\sum_{i = 1}^k (1 - d_i) \varepsilon_i(0)}{\sum_{i = 1}^k (1 - d_i)}\bigg|= O_\mathbb{P}(u_n).
\end{align}
Apply Equation (A.4.2) in \cite{csorgo1997limit} for the partial sum with at least $n - s_n$ terms and using $\max_{1 \leq k \leq s_n}|\frac{1}{n - k}\sum_{i = k+1}^n (d_i - \xi)| = o_\mathbb{P}(1)$, we have
\begin{align}\label{eq: imbalanced reg partial sum large}
\nonumber \max_{1 \leq k \leq s_n} \sqrt{k} \cdot \bigg|\frac{\sum_{i = k+1}^n d_i\varepsilon_i(1)}{\sum_{i = k+1}^n d_i}\bigg|
& = \max_{1 \leq k \leq s_n} \sqrt{k} \cdot \bigg|\frac{n - k}{\sum_{i = k+1}^n d_i} \bigg|\cdot \bigg|\frac{1}{n - k} \sum_{i = k+1}^n d_i \varepsilon_i(1)\bigg| \\
\nonumber & \leq \sqrt{\frac{s_n}{n - s_n}} \bigg(\xi + \min_{1 \leq k \leq s_n}\frac{1}{n - k}\sum_{i = k+1}^n (d_i - \xi)\bigg)^{-1} \cdot \max_{1 \leq k \leq s_n}\bigg|\frac{1}{\sqrt{n - k}} \sum_{i = k+1}^n d_i \varepsilon_i(1)\bigg| \\
& = O_\mathbb{P} (v_n).
\end{align}
The same bound hold for $\max_{1 \leq k \leq s_n} \sqrt{k} \cdot |\frac{\sum_{i = k+1}^n (1 - d_i)\varepsilon_i(1)}{\sum_{i = k+1}^n (1 - d_i)}|$ by a similar argument. Putting together Equations~\eqref{eq: imbalanced ipw partial sum},\eqref{eq: imbalanced ipw partial sum large}, \eqref{eq: imbalanced reg partial sum}, \eqref{eq: imbalanced reg partial sum large}, we have
\begin{align*}
\max_{\ell = 1,2}\max_{1 \leq k \leq s_n} \sqrt{k} |R_\ell(k)| = O_\mathbb{P} (u_n + v_n).
\end{align*}
From the decomposition in Equation~\eqref{eq: decomp} and the symmetry for $k \in [1,s_n]$ and $k \in [n - s_n,n]$, the conclusion follows.
\subsection{Proof of Theorem~\ref{sa-thm: imbalance reg}}
We break down the proofs into two steps.
\begin{center}
\textbf{Step 1: Approximation of reg-score by ipw-score}
\end{center}
Let $0 < a < b < 1$. Let $\rho_n$ be a sequence of real numbers taking values in $(0,1)$ to be determined, and take $s_n = \exp((\log n)^{\rho_n})$. Then for large enough $n$, we have $s_n \leq n^a \leq n^b \leq n - s_n$. Consider the event $A_n := \{\exists \ell \in [p]: \max_{k \in [n]} \mathscr{I}^\mathtt{DIM}(k,\ell) > \max_{k \notin [s_n,n - s_n]} \mathscr{I}^\mathtt{DIM}(k,\ell)\}$. By Equation (A.4.18) from \cite{csorgo1997limit},
\begin{align*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} = O_\mathbb{P}(\sqrt{\rho_n \log \log(n)}).
\end{align*}
Then controlling the difference between $\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)$ and $\mathscr{I}^{\mathtt{DIM}}(k,\ell)$ by Lemma~\ref{lem: approximation -- imbalanced region},
\begin{align}\label{eq: reg small}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n}\mathscr{I}^{\mathtt{DIM}}(k,\ell) = O_\mathbb{P}\bigg(\rho_n \log \log (n) + \frac{s_n}{n - s_n} \log \log(n)\bigg)
\end{align}
By Lemma~\ref{lem: approximation -- balanced region} with the choice $r_n = s_n$,
\begin{align*}
\max_{s_n < k < n - s_n}\sqrt{\mathscr{I}^{\mathtt{DIM}}(k,\ell)} & = \max_{s_n < k < n - s_n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} + O_\mathbb{P} \bigg(\frac{\log \log (n)^{1/2}}{s_n^{1/4}}\bigg) \\
& \geq \max_{1 \leq k \leq n } \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} - \max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} + O_\mathbb{P} \bigg(\frac{\log \log (n)^{1/2}}{s_n^{1/4}}\bigg).
\end{align*}
Equation (A.4.20) in \cite{csorgo1997limit} imply that $(2 \log \log (n))^{-1/2} \max_{1 \leq k \leq n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} = 1 +o_{\mathbb{P}}(1)$ and $(2 \log \log (n))^{-1/2}\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} = \rho_n(1 + o_\mathbb{P}(1))$. Hence
\begin{align}\label{eq: reg large}
\max_{1 \leq k \leq n} \sqrt{\bar\mathscr{I}^{\mathtt{IPW}}(k,\ell)} \geq \sqrt{2 \log \log (n)} + O_\mathbb{P}(\sqrt{\rho_n \log \log (n)}) + O_\mathbb{P} \bigg(\frac{\log \log (n)^{1/2}}{s_n^{1/4}}\bigg)
\end{align}
Choose $\log \log \log \log(n)/\log \log(n) \ll \rho_n \ll 1$, then by Equation~\eqref{eq: reg small} and \eqref{eq: reg large},
\begin{align*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n}\mathscr{I}^{\mathtt{DIM}}(k,\ell) = o_\mathbb{P}(\log \log (n)), \text{ and } \max_{s_n \leq k \leq n - s_n}\mathscr{I}^{\mathtt{DIM}}(k,\ell) = \sqrt{2 \log \log (n)}(1 + o_\mathbb{P}(1)).
\end{align*}
Hence
\begin{align*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n}\mathscr{I}^{\mathtt{DIM}}(k,\ell) = o_\mathbb{P} \bigg(\max_{s_n \leq k \leq n - s_n}\mathscr{I}^{\mathtt{DIM}}(k,\ell)\bigg), \qquad \ell \in [p],
\end{align*}
which by a union bound implies
\begin{align*}
\limsup_{n \rightarrow \infty} \mathbb{P}(A_n) = 0.
\end{align*}
Observe that on the event $A_n^c$, the argmax for $\mathscr{I}^{\mathtt{DIM}}$ should be inside $[s_n, n - s_n]$. Hence
\begin{align*}
& \mathbb{P}\Big(\exists \ell \in [p]: \max_k\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}^{\mathtt{DIM}}(k,j), \; \max_{k}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k \notin [n^a,n^b]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \Big) \\
& \geq \mathbb{P}\Big(\exists \ell \in [p]: \max_k\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}^{\mathtt{DIM}}(k,j), \; \max_{k}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k \notin [n^a,n^b]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \text{ and } A_n^c \Big) - \mathbb{P}(A_n) \\
& \geq \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j), \\
& \qquad \qquad \qquad \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k \notin [n^a,n^b], k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \Big) - 2 \mathbb{P}(A_n).
\end{align*}
Now we focus on the first term. By symmetry in the $p$ coordinates,
\begin{align*}
& \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \Big) \\
& = p \mathbb{P}\Big(\max_{k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,1) > \max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\mathtt{DIM}}(k,1) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,1) \Big) \\
& \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j) < z, \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\mathtt{DIM}}(k,1) > z > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,1) \Big) \\
& \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j) < z, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,1) < z\Big) \\
& \qquad \qquad - p \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j) < z, \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,1) > z\Big).
\end{align*}
Then using the fact that $\bar \mathscr{I}^\mathtt{IPW}(k,\ell)$ approximates $\mathscr{I}^\mathtt{DIM}(k,\ell)$ from Lemma~\ref{lem: approximation -- balanced region}, we have
\begin{align*}
& \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \Big) \\
& \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z - v_n, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z - v_n\Big) \\
& \qquad \qquad - p \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z + v_n, \max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z - v_n\Big),
\end{align*}
where $v_n = O_\mathbb{P}(\log \log (n) s_n^{-1/2})$.
\begin{center}
\textbf{Step 2: Ipw-score approximation by Gaussian approximation}
\end{center}
Observe that the choice $s_n = \exp(\log(n)^{\rho_n})$ for $\log \log \log \log(n)/\log \log(n) \ll \rho_n \ll 1$ implies $\upsilon_n = o_{\mathbb{P}}((\log \log (n))^{-1/2})$. Let $\epsilon > 0$. Then
\begin{align*}
& \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z - v_n, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z - v_n\Big) \\
& \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}} \bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z + v_n, \max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z - v_n \Big) \\
& \geq \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z - \frac{\epsilon}{\sqrt{2 \log \log(n)}}, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z - \frac{\epsilon}{\sqrt{2\log \log (n)}} \Big)\\
& \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}} \bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z + \frac{\epsilon}{\sqrt{2 \log \log(n)}}, \max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z - \frac{\epsilon}{\sqrt{2\log \log (n)}} \Big) \\
& \qquad - \mathbb{P}(|v_n| > \frac{\epsilon}{\sqrt{2\log \log n}}).
\end{align*}
Choosing $z_n(u) = \frac{2 \log \log (n) + 1/2 \log \log \log (n) + u - 1/2 \log(\pi)}{\sqrt{2 \log \log (n)}}$, and from the proof of Theorem~\ref{sa-thm:master}, we have
\begin{align*}
& \liminf_{n \rightarrow \infty }\sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z - \frac{\epsilon}{\sqrt{2 \log \log n}}, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big)\\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}} \bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z + \frac{\epsilon}{\sqrt{2 \log \log n}}, \max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big) \\
& \qquad \qquad - \mathbb{P}(|v_n| > \frac{\epsilon}{\sqrt{2 \log \log n}}) \\
& \geq \liminf_{n \rightarrow \infty } \sup_{u \in \mathbb{R}}\mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}}, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big)\\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}} \bar \mathscr{I}^{\mathtt{IPW}}(k,j) < z_n(u) + \frac{\epsilon}{\sqrt{2 \log \log n}}, \max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big) \\
& \qquad \qquad - \mathbb{P}(|v_n| > \frac{\epsilon}{\sqrt{2 \log \log n}}), \\
& \geq \liminf_{n \rightarrow \infty } \sup_{u \in \mathbb{R}}\mathbb{P}\Big(\max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big)^{p-1} \mathbb{P} \Big(\max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big)\\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}} \bar \mathscr{I}^{\mathtt{IPW}}(k,1) < z_n(u) + \frac{\epsilon}{\sqrt{2 \log \log n}}\Big)^{p-1} \mathbb{P} \Big(\max_{k \in [s_n, n - s_n]}\bar \mathscr{I}^{\mathtt{IPW}}(k,1) > z_n(u) - \frac{\epsilon}{\sqrt{2 \log \log n}} \Big) \\
& \geq \sup_{u \in \mathbb{R}} \exp\Big(-2(p-1)e^{-(u - \epsilon -\log(2))}\Big) \exp\Big(-2e^{-(u - \epsilon -\log(2-(b-a)))}\Big) \\
& \qquad \qquad - \exp\Big(-2(p-1)e^{-(u + \epsilon -\log(2))}\Big) \exp\Big(-2e^{-(u - \epsilon -\log(2))}\Big).
\end{align*}
Now let $\epsilon \downarrow 0$, and then all previous steps together implies
\begin{align*}
& \liminf_{n \rightarrow \infty} \mathbb{P}\Big(\exists \ell \in [p]: \max_k\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}^{\mathtt{DIM}}(k,j), \; \max_{k}\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k \notin [n^a,n^b]}\mathscr{I}^{\mathtt{DIM}}(k,\ell) \Big) \\
& \geq \sup_{u \in \mathbb{R}} \exp\Big(-2(p-1)e^{-(u -\log(2))}\Big) \Big(\exp\Big(-2e^{-(u -\log(2-(b-a)))}\Big) -
\exp\Big(-2e^{-(u -\log(2))}\Big)\Big) \\
& \geq \frac{b - a}{2 e}.
\end{align*}
\subsection{Proof of Theorem~\ref{sa-thm:rates_reg}}
The proofs follow the essentially same logic as the proof for Theorem~\ref{sa-thm:rates}, with some tricks for the random numerator in $\frac{\sum_{1 \leq i \leq k} d_i \varepsilon_i(1)}{\sum_{1 \leq i \leq k} d_i}$.
\begin{center}
\textbf{Part 1: Inconsistency for Uniform Estimation Rates}
\end{center}
Denote the optimal index for splitting based on the $\ell$'s coordinate by
\begin{align*}
\hat{\imath}_{\text{DIM},\ell} = \operatorname*{arg\,max}_{k \in [n]} \bar\mathscr{I}^{\mathtt{DIM}}(k,\ell), \qquad \ell \in [p].
\end{align*}
For notational simplicity, denote
\begin{align*}
\bar \tau^\mathtt{DIM}_L(k, \ell) & = \tau^\mathtt{DIM}_L(k, \ell) - \tau = \frac{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)} \varepsilon_{\pi_{\ell}(i)}(1)}{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)}} - \frac{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)}) \varepsilon_{\pi_{\ell}(i)}(0)}{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)})}, \\
\bar \tau_L^\mathtt{DIM}(\ell) & = \bar \tau_L^\mathtt{DIM}(\hat{\imath}_{\text{DIM},\ell},\ell),\\
\bar \tau^\mathtt{DIM}_R(k,\ell) & = \tau^\mathtt{DIM}_R(k,\ell) - \tau = \frac{\sum_{k < i \leq n}d_{\pi_{\ell}(i)} \varepsilon_{\pi_{\ell}(i)}(1)}{\sum_{k < i \leq n}d_{\pi_{\ell}(i)}} - \frac{\sum_{k < i \leq n}(1 - d_{\pi_{\ell}(i)}) \varepsilon_{\pi_{\ell}(i)}(0)}{\sum_{k < i \leq n}(1 - d_{\pi_{\ell}(i)})}, \\
\bar \tau_R^\mathtt{DIM}(\ell) & = \bar \tau_R^\mathtt{DIM}(\hat{\imath}_{\text{DIM},\ell},\ell),
\end{align*}
and consider the event
\begin{align*}
\mathtt{Imblce}_{\ell}^\mathtt{DIM} = \{\max_k\bar\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k,j\neq \ell}\bar\mathscr{I}^{\mathtt{DIM}}(k,j), \; \max_{k}\bar\mathscr{I}^{\mathtt{DIM}}(k,\ell) > \max_{k \notin [n^b, n - n^b]}\bar\mathscr{I}^{\mathtt{DIM}}(k,\ell)\}, \qquad \ell \in [p].
\end{align*}
Since we assume $\mu_0 \equiv c_0$ and $\mu_1 \equiv c_1$ with $c_1 - c_0 = \tau$, we have on $\mathtt{Imblce}_{\ell}^\mathtt{DIM} \cap \{\hat{\iota}_{\ell}^\mathtt{DIM} \leq n/2\}$,
\begin{align}\label{eq: rate reg -- decompose}
& \nonumber \sup_{x \in \mathcal{X}}|\hat\tau(x) - \tau|^2
\geq \bar \tau^\mathtt{DIM}_L(\ell)^2 \\
& \geq \frac{1}{\min\{\hat{\imath}_{\text{DIM},\ell}, n - \hat{\imath}_{\text{DIM},\ell}\}} \bigg(\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 - (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} \leq n/2) \bigg).
\end{align}
Take $\bar\tau^{\mathtt{DIM}} = \frac{\hat{\imath}_{\text{DIM},\ell}}{n} \bar\tau^\mathtt{DIM}_L + \frac{n - \hat{\imath}_{\text{DIM},\ell}}{n} \bar\tau^\mathtt{DIM}_R$. Then
\begin{align*}
\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2
& \geq \hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 - n \bar\tau^\mathtt{DIM} \\
& = \frac{\hat{\imath}_{\text{DIM},\ell}(n - \hat{\imath}_{\text{DIM},\ell})}{n}\bigg(\bar\tau_L^\mathtt{DIM} - \bar \tau_R^\mathtt{DIM}\bigg)^2
\end{align*}
By Lemma~\ref{lem: approximation -- balanced region} and Lemma~\ref{lem: approximation -- imbalanced region} with $r_n = s_n = \exp((\log n)^{\rho_n})$ for $\log \log \log \log (n)/\log \log(n) \ll \rho_n \ll 1$,
\begin{align*}
\frac{\hat{\imath}_{\text{DIM},\ell}(n - \hat{\imath}_{\text{DIM},\ell})}{n}\bigg(\bar\tau_L^\mathtt{DIM} - \bar \tau_R^\mathtt{DIM}\bigg)^2
& = \frac{\hat{\iota}^{\text{ipw}}(n - \hat{\iota}^{\text{ipw}})}{n}\bigg(\bar\tau_L^\mathtt{IPW} - \bar \tau_R^\mathtt{IPW}\bigg)^2 + o_\mathbb{P}(\log \log (n)) \\
& = \max_{1 \leq k \leq n} \bar \mathscr{I}^\mathtt{IPW}(k) + o_\mathbb{P}(\log \log n).
\end{align*}
By Theorem A.4.1 in \cite{csorgo1997limit}, $\max_{1 \leq k \leq n} \bar \mathscr{I}^\mathtt{IPW}(k) = 2 \log \log (n) (1 + o_\mathbb{P}(1))$.
Moreover,
\begin{align*}
\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} > n/2) & \leq \max_{k > n/2} k \cdot \bigg(\frac{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)} \varepsilon_{\pi_{\ell}(i)}(1)}{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)}} - \frac{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)}) \varepsilon_{\pi_{\ell}(i)}(0)}{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)})} \bigg)^2 \\
& \leq \max_{k > n/2} 2 k \cdot \bigg(\frac{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)} \varepsilon_{\pi_{\ell}(i)}(1)}{\sum_{1 \leq i \leq k}d_{\pi_{\ell}(i)}}\bigg)^2 + 2k \cdot \bigg(\frac{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)}) \varepsilon_{\pi_{\ell}(i)}(0)}{\sum_{1 \leq i \leq k}(1 - d_{\pi_{\ell}(i)})} \bigg)^2.
\end{align*}
For simplicity in showing the upper bound for $\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} > n/2) $, we assume $\pi$ is the identity permulation. Take $b_i = \sum_{1 \leq j \leq i} d_j$. By Equation (8) from \cite{shorack1976inequalities}, for any $\lambda > 0$,
\begin{align*}
\mathbb{P} \bigg(\max_{k > n/2} \Big|\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\Big| \geq \lambda \bigg| (d_i)_{1 \leq i \leq n}\bigg) & \leq 16 \sum_{i > n/2} \frac{d_i\mathbb{V}[\varepsilon_i(1)]}{b_i^2} \lambda^{-2} \\
& \leq 16 \sum_{i > b_{n/2}}\frac{1}{i^2} \lambda^{-2} \mathbb{V}[\varepsilon_i(1)] \\
& \leq \frac{8}{3}\pi^2 \lambda^{-2} \mathbb{V}[\varepsilon_i(1)] \frac{1}{b_{n/2}},
\end{align*}
And since $d_i$'s are i.i.d with $\mathbb{E}[d_i] = \xi > 0$, we have
\begin{align*}
(b_{n/2})^{-1} = (n/2)^{-1} \Big(\xi + \frac{2}{n}\sum_{i = 1}^{n/2}(d_i - \xi)\Big)^{-1} = O_\mathbb{P}(n^{-1}).
\end{align*}
Hence uncondition on $(d_i)_{1 \leq i \leq n}$, and we have
\begin{align*}
\max_{k \geq n/2} k \cdot \bigg(\frac{\sum_{i = 1}^k d_i \varepsilon_i(1)}{\sum_{i =1}^k d_i}\bigg)^2 = O_\mathbb{P}(1) = o_\mathbb{P}(\log \log (n)).
\end{align*}
By a similar term for control, and a symmetric argument for the right node,
\begin{align*}
\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} > n/2) + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} \leq n/2) = o_\mathbb{P}(\log \log (n)).
\end{align*}
Fix $\epsilon > 0$. Consider the events
\begin{align*}
A_{\ell}^{\epsilon} & = \bigg\{\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 \geq (2 - \epsilon) \log \log(n)\bigg\}, \\
B_{\ell}^{\epsilon} & = \bigg\{\hat{\imath}_{\text{DIM},\ell} \bar{\tau}_L^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} > n/2) + (n - \hat{\imath}_{\text{DIM},\ell}) \bar{\tau}_R^{\mathtt{DIM}}(\ell)^2 \mathds{1}(\hat{\imath}_{\text{DIM},\ell} \leq n/2) \leq 2 \epsilon \log \log (n)\bigg\}.
\end{align*}
The above arguments show that $\liminf_{n \rightarrow \infty} \mathbb{P}(A_{\ell}^{\epsilon}) = \liminf_{n \rightarrow \infty} \mathbb{P}(B_{\ell}^{\epsilon}) = 1$. From Theorem~\ref{sa-thm: imbalance reg}, $$\mathbb{P}(\mathtt{Imblce}_{\ell}^\mathtt{DIM}) \geq \frac{b}{pe}.$$ It then follows from a union bound argument that
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\bigg(\sup_{\mathbf{x}\in\mathcal{X}}|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\bigg) \geq \frac{b}{e}.
\end{equation*}
\begin{center}
\textbf{Part 2: Inconsistency for Points near the Boundary}
\end{center}
Fix $\mathbf{z} \in \mathcal{X}$ such that $z_{\ell} \leq n^{a - 1}$. Since the order statistics $x_{(n^a),\ell} = n^{a - 1}(1 + o_\mathbb{P}(1))$, on the event $n^a \leq \hat{\imath}_{\text{DIM},\ell} \leq n^b$, if $z_{\ell} \leq (1 + o_\mathbb{P}(1))n^{a-1}$, then $z_{\ell} \leq x_{(n^a)} \leq x_{(\hat{\imath}_{\text{DIM},\ell}),\ell}$, and on the event $\mathtt{Imblce}_{\ell}^\mathtt{DIM}$,
\begin{align*}
|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{z}) - \tau|^2
= \bar \tau_L^\mathtt{DIM}(\ell)^2
& \geq \frac{1}{\hat{\imath}_{\text{DIM},\ell}} \bigg(\hat{\imath}_{\text{DIM},\ell} \bar \tau_L^\mathtt{DIM}(\ell)^2 + (n - \hat{\imath}_{\text{DIM},\ell}) \bar \tau_R^\mathtt{DIM}(\ell)^2 - (n - \hat{\imath}_{\text{DIM},\ell})\bar \tau_R^\mathtt{DIM}(\ell)^2\bigg) \\
& \geq \frac{1}{\hat{\imath}_{\text{DIM},\ell}} \bigg(\max_{1 \leq k \leq n} k \bar \tau_L^\mathtt{DIM}(k,\ell)^2 + (n - k) \bar \tau_R^\mathtt{DIM}(k,\ell)^2 - \max_{k \leq n^b}(n - k) \bar \tau_R^\mathtt{DIM}(k,\ell)^2\bigg) \\
& \geq \frac{(2 + o_\mathbb{P}(1)) \log \log (n)}{\hat{\imath}_{\text{DIM},\ell}} \\
& \geq \frac{(2 + o_\mathbb{P}(1)) \log \log (n)}{n^b},
\end{align*}
where the second to last line is due to a similar argument as in the proof of part 1. By a symmetry argument for the event $\{n - n^b \leq \hat{\imath}_{\text{DIM},\ell} \leq n - n^a\}$, we have
\begin{equation*}
\liminf_{n\to\infty} \inf_{\mathbf{x}\in \mathcal{X}_n} \mathbb{P}\Big(|\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}(\mathbf{x}) - \tau| \geq \sigma n^{-b/2}\sqrt{(2+o(1))\log\log(n)}\Big) \geq \frac{b-a}{2e},
\end{equation*}
where $\mathcal{X}_n = \{\mathbf{x} \in [0,1]^p: x_j = o(1) n^{a-1} \text{ or } 1 - x_j = o(1) n^{a-1} \text{ for some } j \in [p]\}$, and $\sigma^2 = \mathbb{V}[\frac{d_i y_i(1)}{\xi} + \frac{(1 - d_i) y_i(0)}{1 - \xi}]$.
\subsection{Proof of Theorem~\ref{sa-thm: uniform minimax rates regression}}
Due to the recursive splitting and Theorem~\ref{sa-thm: imbalance reg}, the optimal split index $\hat \imath_\mathtt{DIM}$ at the $k$-th split ($k \geq 1$) also satisfies
\begin{align*}
\liminf_{n\to\infty} \mathbb{P}\big( \hat\imath_\mathtt{DIM} \leq n^{b}\big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath_\mathtt{DIM} \big) \geq \frac{b}{2 e}.
\end{align*}
Hence the same argument as Part 1 in the proof of Theorem~\ref{sa-thm:rates_reg} leads to the result.
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency NSS reg}}
For notational simplicity, denote $\hat\tau^{\mathtt{NSS}}_{\mathtt{DIM}}$ by $\hat{\tau}$, the data-driven partition $\mathcal{D}_{\mathsf{T}}$ by $\mathcal{P}$.
\subsubsection*{Reduction to least square prediction error.}
Observe that the leaf nodes value coincide with a least square projection given $\mathcal{P}$: For $\mathsf{t} \in \mathcal{P}$, we have $\hat{\tau}(\mathsf{t}) = \hat{b}_{\mathsf{t}}$, where
\begin{align*}
\hat{a}_{\mathsf{t}}, \hat{b}_{\mathsf{t}} =
\begin{cases}
\operatorname*{arg\,min}_{a, b} \sum_{i = 1}^n \mathds{1}(\mathbf{x}_i \in \mathsf{t}) (y_i - a - b \; d_i)^2 & \text{if} \sum_{i = 1}^n \mathds{1}(\mathbf{x}_i \in \mathsf{t}) > 0, \\
0, 0 & \text{otherwise}.
\end{cases}
\end{align*}
Consider the outcome prediction model based on partition $\mathcal{P}$:
\begin{align}\label{sa-eq: dim least square representation}
\nonumber \hat{g}(\mathbf{x}, d) & = \sum_{\mathsf{t} \in \mathcal{P}} \mathds{1}(\mathbf{x} \in \mathsf{t}) (\hat{a}_{\mathsf{t}} + \hat{b}_{\mathsf{t}} d) \\
& = \hat{A}(\mathbf{x}) + \hat{B}(\mathbf{x}) \; d,
\end{align}
where
\begin{align*}
\hat{A}(\mathbf{x}) = \sum_{\mathsf{t} \in \mathcal{P}} \mathds{1}(\mathbf{x} \in \mathsf{t}) \hat{a}_{\mathsf{t}}, \qquad
\hat{B}(\mathbf{x}) = \sum_{\mathsf{t} \in \mathcal{P}} \mathds{1}(\mathbf{x} \in \mathsf{t}) \hat{b}_{\mathsf{t}}.
\end{align*}
First, we show that for $L_2$-consistency of treatment effect estimation, it is enough to look at the $L_2$ loss for outcome prediction. Denote by $P_{X,d}$ the joint distribution of $(\mathbf{x}_i, d_i)$. Since we assumed $\mathbf{x}_i$ and $d_i$ are independent, we have $P_{X,d} = P_X \times P_d$, where $P_X$ and $P_d$ are the marginal distributions of $X$ and $d$. Given Assumption~\ref{sa-assump: dgp-causal}, the target outcome prediction model is
\begin{align*}
g^{\ast}(\mathbf{x}_i, d_i) = \mathbb{E}[y_i|\mathbf{x}_i, d_i] = \mu + \tau \; d_i, \qquad \mu = \mathbb{E}[y_i(0)], \quad \tau = \mathbb{E}[y_i(1) - y_i(0)].
\end{align*}
Hence
\begin{align}\label{sa-eq: tau hat to g hat}
\nonumber & \mathbb{E} [\lVert \hat{g} - g^{\ast} \rVert^2] \\
\nonumber & = \mathbb{E} \bigg[\int_{\mathcal{X} \times \{0,1\}} (\hat{g}(\mathbf{x},d) - \mu - \tau x)^2 d P_{X,d}(\mathbf{x},d)\bigg] \\
\nonumber & = \mathbb{E} \bigg[\int_{\mathcal{X} \times \{0,1\}} (\hat{A}(\mathbf{x}) + \hat{B}(\mathbf{x}) d - \mu - \tau d)^2 d P_X(\mathbf{x}) \times P_d(d)\bigg] \\
\nonumber & = \mathbb{E} \bigg[\int_{\mathcal{X} \times \{0,1\}} ( d \;(\hat{A}(\mathbf{x}) + \hat{B}(\mathbf{x}) - \mu - \tau) + (1 - d) \; (\hat{A}(\mathbf{x}) - \mu))^2 d P_X(\mathbf{x}) \times P_d(d)\bigg] \\
\nonumber & = \mathbb{E} \bigg[\int_{\mathcal{X} \times \{0,1\}} d \; (\hat{A}(\mathbf{x}) + \hat{B}(\mathbf{x}) - \mu - \tau)^2 + (1 - d) \; (\hat{A}(\mathbf{x}) - \mu)^2 d P_X(\mathbf{x}) \times P_d(d)\bigg] \\
\nonumber & = \mathbb{E} \bigg[ \xi \int_{\mathcal{X}} \; (\hat{A}(\mathbf{x}) + \hat{B}(\mathbf{x}) - \mu - \tau)^2 d P_X(\mathbf{x}) + (1 - \xi) \int_{\mathcal{X}} \; (\hat{A}(\mathbf{x}) - \mu)^2 d P_X(\mathbf{x}) \bigg] \\
& = \xi \mathbb{E}[\lVert \hat{A} + \hat{B} - \mu - \tau \rVert^2] + (1 - \xi) \mathbb{E}[\lVert \hat{A} - \mu \rVert^2].
\end{align}
It follows that
\begin{align*}
& \mathbb{E} [\lVert \hat{\tau} - \tau \rVert^2]
= \mathbb{E} [\lVert \hat{B} - \tau \rVert^2 ]
\leq \frac{4}{\min\{\xi, 1 - \xi\}} \mathbb{E} [\lVert \hat{g} - g^{\ast} \rVert^2].
\end{align*}
\subsubsection*{Error Bound for Least Square Prediction.}
Now, we bound the least square error $\mathbb{E}[\lVert \hat{g} - g^{\ast}\rVert^2]$ following the strategy for \cite[Theorem 4.3]{klusowski2024large}. First, assume $|y_i(t)| \leq U$, $i = 1,2, \cdots, n$, $t = 0,1$, for some $U \geq 0$. Decompose by
\begin{align*}
\lVert \hat{g} - g^{\ast}\rVert^2 = E_1 + E_2,
\end{align*}
where
\begin{align*}
E_1 & = \lVert \hat{g} - g^{\ast}\rVert^2 - 2 (\lVert y - \hat{g} \rVert_{\mathcal{D}}^2 - \lVert y - g^{\ast}\rVert_{\mathcal{D}}^2) - \alpha - \beta, \\
E_2 & = 2 (\lVert y - \hat{g} \rVert_{\mathcal{D}}^2 - \lVert y - g^{\ast}\rVert_{\mathcal{D}}^2) + \alpha + \beta.
\end{align*}
The least square representation \eqref{sa-eq: dim least square representation} implies that
\begin{align}\label{sa-eq: empirical risk minimization}
\lVert y - \hat{g}\rVert_{\mathcal{D}}^2
\leq \min_{a \in \mathbb{R}, b \in \mathbb{R}} \sum_{i = 1}^n (y_i - a - b\; d_i)^2
\leq \lVert y - \mu - \tau d\rVert_{\mathcal{D}}^2
= \lVert y - g^{\ast}\rVert_{\mathcal{D}}^2,
\end{align}
which implies
\begin{align*}
E_2 \leq \alpha + \beta.
\end{align*}
We control $E_1$ using uniform law of large number arguments. Notice that $\hat{g}$ is one member of the class $\mathscr{G}_{n} = \{A(\mathbf{x}) + d \; B(\mathbf{x}): A, B \in \mathscr{H}_n\}$, where $\mathscr{H}_n$ is the class of piecewise constant functions (bounded by $U$) on partitions $\mathbb{P} \in \Pi_n$. Here
\begin{align*}
\Pi_n = \{\mathcal{P}(\{(\mathbf{x}_1,d_1,y_1), \cdots, (\mathbf{x}_n, d_n, y_n)\}): (\mathbf{x}_i, d_i, y_i) \in \mathbb{R}^p \times \mathbb{R} \times \mathbb{R}\},
\end{align*}
is the family of all achievable partitions $\mathcal{P}$ by growing a depth $K$ binary tree on $n$ points by iteratively splitting in $\mathbf{x}$-space based on any criterion. By \cite[Equation B.33]{klusowski2024large},
\begin{align*}
N \bigg(\frac{\beta}{40 U}, \mathscr{H}_n, \lVert \cdot \rVert_{P_{X^n}, 1} \bigg) \leq (n p)^{2^K} \bigg(\frac{417 e U^2}{\beta}\bigg)^{2^{K+1}}.
\end{align*}
A union bound then gives
\begin{align*}
N \bigg(\frac{\beta}{80 U}, \mathscr{G}_n, \lVert \cdot \rVert_{P_{X^n}, 1} \bigg) \leq 2 (n p)^{2^K} \bigg(\frac{417 e U^2}{\beta}\bigg)^{2^{K+1}},
\end{align*}
where $P_{X^n}$ is the empirical measure based on $X^n = (X_1, \cdots, X_n)$, $X_i \in \mathbb{R}^p$ for all $i$. Since $\hat{g} \in \mathscr{G}_n$, we can then use \cite[Theorem 11.4]{Gyorfi-etal_2002_book} to get
\begin{align*}
\mathbb{P}(E_1 \geq 0) & \leq \mathbb{P}(\exists g \in \mathscr{G}_n: \lVert \hat{g} - g^{\ast}\rVert^2 \geq 2 (\lVert y - \hat{g} \rVert_{\mathcal{D}}^2 - \lVert y - g^{\ast}\rVert_{\mathcal{D}}^2) + \alpha + \beta) \\
& \leq 14 \sup_{X^n} N \bigg(\frac{\beta}{80 U}, \mathscr{G}_n, \lVert \cdot \rVert_{P_{X^n}, 1} \bigg) \exp \bigg(- \frac{\alpha n}{2568 U^4}\bigg) \\
& \leq 28 (n p)^{2^K} \bigg(\frac{417 e U^2}{\beta}\bigg)^{2^{K+1}} \exp \bigg(- \frac{\alpha n}{2568 U^4}\bigg).
\end{align*}
Choosing $\alpha \propto \frac{U^4 2^K \log(np)}{n}$, and $\beta \propto \frac{U^2}{n}$, then we have
\begin{align*}
\mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2] \leq C \bigg(\frac{U^4 2^K \log(np)}{n} + \frac{U^2}{n} \bigg),
\end{align*}
where $C$ is a positive universal constant.
Now we relax the condition that $|y_i(t)| \leq U$. Take $A = \{|y_i(t)| \leq U, \forall i = 1, \cdots, n, t = 0,1\}$. Then
\begin{align}\label{sa-eq: truncation}
\nonumber \mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2]
& = \mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2 \mathds{1}(A)] + \mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2 \mathds{1}(A^c)] \\
& \leq C \bigg(\frac{U^4 2^K \log(np)}{n} + \frac{U^2}{n} \bigg) + \mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2 \mathds{1}(A^c)].
\end{align}
A union bound gives
\begin{align*}
\mathbb{P}(A^c) & \leq n \mathbb{P}(|y_i(0)| \geq U) + n \mathbb{P}(|y_i(1)| \geq U) \\
& \leq n \exp (- |U - \mu_0| ) + n \exp(-|U - \mu_1|).
\end{align*}
Using Cauchy-Schwarz inequality,
\begin{align*}
\mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2 \mathds{1}(A^c)]
& \leq \sqrt{\mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^4]\mathbb{P}(A^c)} \\
& \leq \sqrt{8 n \max_{t = 0,1}(\mu_t^4 + \mathbb{E}[\varepsilon_i(t)^4])} \; n \max_{t = 0,1}\exp (- |U - \mu_t| ).
\end{align*}
Choosing $U = \max\{\mu_0, \mu_1\} + 4 \log(n)$, we have
\begin{align*}
\mathbb{E}[ \lVert \hat{g} - g^{\ast}\rVert^2 \mathds{1}(A^c)] \leq \frac{C}{n},
\end{align*}
for some absolute constant $C$. Putting it back to Equation~\eqref{sa-eq: truncation}, we get the desired conclusion.
For the high probability bound, the same analysis as Equation~\eqref{sa-eq: tau hat to g hat} in almost sure sense gives
\begin{align*}
\lVert \hat{\tau} - \tau \rVert^2
& \leq \mathbb{E} [\lVert \hat{B} - \tau \rVert^2 ]
\leq \frac{4}{\min\{\xi, 1 - \xi\}} \lVert \hat{g} - g^{\ast} \rVert^2,
\end{align*}
almost surely. Using sub-exponentianity of $\varepsilon_i(t)$,
\begin{align*}
\lVert g - g^{\ast}\rVert \mathds{1}(A)
& \leq E_1 \mathds{1}(A) + E_2 \mathds{1}(A) \\
& \leq C_1 \bigg(\frac{U^4 2^K \log(np)}{n} + \frac{U^2}{n} \bigg),
\end{align*}
with probability at least $n^{-C_2}$, where $C_1$ and $C_2$ are some positive absolute constants. Sub-exponentianity of $\varepsilon_i(t)$, $1 \leq i \leq n$, $t = 0,1$, implies that $\mathbb{P}(A^c) = n^{-C_3}$ if we choose $U = C_4 \log(n)$, where $C_3$ and $C_4$ are positive constants only depending on the distribution of $(\varepsilon_i(0), \varepsilon_i(1))$. Combining with the previous two inequalities, we get the second conclusion.
\subsection{Proof of Theorem~\ref{sa-coro: honest output reg}}
Recall $(\hat \imath, \hat \jmath)$ denotes the optimal splitting index and coordinate for the decision stump. Denote $\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}(\mathbf{x})$ by $\check{\tau}(\mathbf{x})$ for simplicity. We use $(y_i,\mathbf{x}_i^{\top})_{i=1}^M$ to denote $\mathcal{D}_{\mathtt{HON},1}$, which we used to construct the causal tree. Denote by $(\hat \imath, \hat \jmath)$ the splitting index and coordinate at the $K_n$-th step, based on $\mathcal{D}_{\mathtt{HON},1}$.
Use $(\tilde{y}_i,\tilde{\mathbf{x}}_i^{\top})_{i=1}^N$ to denote $\mathcal{D}_{\mathtt{HON},2}$. Then
\begin{align*}
\sup_{\mathbf{x} \in \mathcal{X}} |\check{\tau}(\mathbf{x}) - \tau|
& \geq |\check{\tau}(\mathbf{0}) - \tau| \\
& = \bigg|\frac{\sum_{i = 1}^N \tilde{d}_i \tilde{\varepsilon}_i(1) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N \tilde{d}_i \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} - \frac{\sum_{i = 1}^N (1 - \tilde{d}_i) \tilde{\varepsilon}_i(0) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N (1 - \tilde{d}_i) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} \bigg|.
\end{align*}
Since $(\tilde{\varepsilon}_i(0), \tilde{\varepsilon}_i(1)) \protect\mathpalette{\protect\independenT}{\perp} \tilde{\mathbf{x}}_i$, condition on $\hat{\imath}, \hat{\jmath}$ and $\mathbf{X} = (\mathbf{x}_1, \cdots, \mathbf{x}_n)$, $\tilde{\mathbf{X}} = (\tilde{\mathbf{x}}_1, \cdots, \tilde{\mathbf{x}}_n)$, we have
\begin{align*}
& \frac{\sum_{i = 1}^N \tilde{d}_i \tilde{\varepsilon}_i(1) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N \tilde{d}_i \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} - \frac{\sum_{i = 1}^N (1 - \tilde{d}_i) \tilde{\varepsilon}_i(0) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})}{\sum_{i = 1}^N (1 - \tilde{d}_i) \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath})} \\
& \stackrel{d}{=} \frac{\sum_{i = 1}^{\tilde{\iota}} d_i \varepsilon_i(1) }{\sum_{i = 1}^{\tilde{\iota}} d_i} - \frac{\sum_{i = 1}^{\tilde{\iota}} (1 - d_i) \varepsilon_i(0)}{\sum_{i = 1}^{\tilde{\iota}} (1 - d_i)},
\end{align*}
where
\begin{align*}
\tilde{\imath} = \sum_{i = 1}^N \mathds{1}(\tilde{x}_{i,\hat \jmath} \leq x_{\pi_{\hat \jmath}(\hat \imath), \hat \jmath}).
\end{align*}
Call
\begin{align*}
Z = \Bigl| \frac{\sum_{i = 1}^{\tilde{\iota}} d_i \varepsilon_i(1) }{\sum_{i = 1}^{\tilde{\iota}} d_i} - \frac{\sum_{i = 1}^{\tilde{\iota}} (1 - d_i) \varepsilon_i(0)}{\sum_{i = 1}^{\tilde{\iota}} (1 - d_i)} \Bigr|.
\end{align*}
\paragraph*{High probability lower bound on $Z$.} Denote $n_1 = \sum_{i=1}^{\tilde{\iota}} d_i,
n_0 = \sum_{i=1}^{\tilde{\iota}} (1-d_i),
n_0 + n_1 = \tilde{\iota}.$ And consider the weights
$w_i = \frac{d_i}{n_1} - \frac{1-d_i}{n_0}$, so that $Z = |\sum_{i=1}^{\tilde{\iota}} w_i \varepsilon_i|.$ By Marcinkiewicz–Zygmund inequality and a Jensen's inequality on the square root function, for some absolute constant $c_{\mathrm{MZ}}$,
\begin{align*}
\mathbb{E}[Z \mid \bm D, \tilde{\iota}]
&= \mathbb{E}\Bigl[\Bigl|\sum_{i} w_i \varepsilon_i\Bigr| \Bigm| \bm D, \tilde{\iota} \Bigr] \\
&\ge c_{\mathrm{MZ}} \,
\mathbb{E}\Bigl[\Bigl(\sum_{i} w_i^{2}\varepsilon_i^{2}\Bigr)^{1/2} \Bigm| \bm D, \tilde{\iota} \Bigr] \\
&\ge c_{\mathrm{MZ}} \,
\mathbb{E}\Bigl[
\Bigl(
\sum_{i=1}^{\tilde{\iota}} \frac{d_i}{n_1^{2}} \varepsilon_i(1)^{2}
+ \sum_{i=1}^{\tilde{\iota}} \frac{1-d_i}{n_0^{2}} \varepsilon_i(0)^{2}
\Bigr)^{1/2}
\Bigm| \bm D, \tilde{\iota} \Bigr] \\
&\ge \frac{c_{\mathrm{MZ}}}{n_0 + n_1}
\sum_{i=1}^{\tilde{\iota}}
\Bigl(
d_i \frac{\sqrt{n_0+n_1}}{n_1} \sqrt{\mathbb{E}[\varepsilon_i(1)^{2}]}
+ (1-d_i) \frac{\sqrt{n_0+n_1}}{n_0} \sqrt{\mathbb{E}[\varepsilon_i(0)^{2}]}
\Bigr) \\
&= c_{\mathrm{MZ}} \frac{1}{\sqrt{\tilde{\iota}}}
\Bigl(\sqrt{\mathbb{V}[\varepsilon(1)]} + \sqrt{\mathbb{V}[\varepsilon(0)]}\Bigr).
\end{align*}
Moreover, Assumption~\ref{sa-assump: dgp-causal} implies that
$$
\mathbb{E}[Z^{2} \mid \bm D, \tilde{\iota}]
= \frac{\sum_{i=1}^{\tilde{\iota}} d_i \mathbb{E}[\varepsilon_i(1)^{2}]}{n_1^{2}}
+ \frac{\sum_{i=1}^{\tilde{\iota}} (1-d_i) \mathbb{E}[\varepsilon_i(0)^{2}]}{n_0^{2}}
\ge \left(\frac{1}{n_1} + \frac{1}{n_0}\right) \min\{\mathbb{V}[\varepsilon_i(1)], \mathbb{V}[\varepsilon_i(0)]\}.
$$
The Paley-Zygmund inequality implies for $\theta \in (0,1)$,
\begin{align}\label{sa-eq: Paley-Zygmund}
\nonumber \mathbb{P}\bigl(Z \ge \theta \mathbb{E}[Z \mid \bm D, \tilde{\iota}] \mid \bm D, \tilde{\iota} \bigr)
& \ge (1-\theta^{2}) \frac{\mathbb{E}[Z \mid \bm D, \tilde{\iota}]^{2}}{\mathbb{E}[Z^{2} \mid \bm D, \tilde{\iota}]} \\
& \geq C (1 - \theta^2) \frac{\min\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}}{\max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}} \frac{n_0 n_1}{\tilde{\iota}^2}.
\end{align}
Condition on $\tilde{\iota}$, $n_0 \sim \mathsf{Bernoulli}(\tilde{\iota}, \xi)$. Hence
\begin{align*}
\mathbb{P}\bigl(Z \ge \theta \mathbb{E}[Z \mid \bm D, \tilde{\iota}] \mid \tilde{\iota} \bigr) \geq C (1 - \theta^2) \frac{\min\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}}{\max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}} (\xi - \xi^2)(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0).
\end{align*}
We claim that whenever $s_n \leq \hat{\iota} \leq n - s_n$,
\begin{align}\label{sa-eq: tilde iota inverse}
\mathbb{E} \Bigl[(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0)|\hat{\iota} \Bigr] = 1 + o_{\mathbb{P}}(1).
\end{align}
It then follows from Equation~\eqref{sa-eq: Paley-Zygmund} that whenever $s_n \leq \hat{\iota} \leq M - s_n$,
\begin{align*}
\mathbb{P}\biggl(Z \ge \theta c_{\mathrm{MZ}} \frac{1}{\sqrt{\tilde{\iota}}}
\Bigl(\sqrt{\mathbb{V}[\varepsilon(1)]} + \sqrt{\mathbb{V}[\varepsilon(0)]}\Bigr)\bigg|\hat{\iota}\biggr) \geq C (1 - \theta^2) \frac{\min\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}}{\max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}} (\xi - \xi^2) + o_{\mathbb{P}}(1).
\end{align*}
Choose $\theta = 1/2$, and take
\begin{align*}
\mathtt{C} = C \frac{\min\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}}{\max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}} \frac{\xi - \xi^2}{4}.
\end{align*}
Then by Theorem~\ref{sa-thm: imbalance reg}, we have
\begin{align*}
\mathbb{P}\biggl(Z \ge \frac{1}{2} c_{\mathrm{MZ}} \frac{1}{\sqrt{\tilde{\iota}}}
\Bigl(\sqrt{\mathbb{V}[\varepsilon(1)]} + \sqrt{\mathbb{V}[\varepsilon(0)]}\Bigr), \hat{\iota} \leq n^b \biggr) \geq \mathtt{C}\frac{b}{2e} + o_{\mathbb{P}}(1).
\end{align*}
We can show via the same argument as Theorem~\ref{sa-thm: honest stump} that $\liminf_{n \to \infty}\mathbb{P}(\tilde{\iota} \leq n^b/2|\hat{\iota} \leq n^b) = 1$. Hence
\begin{align*}
\liminf_{n \to \infty} \mathbb{P}\biggl(Z \ge \frac{1}{4} c_{\mathrm{MZ}} \frac{1}{\sqrt{n^b}}
\Bigl(\sqrt{\mathbb{V}[\varepsilon(1)]} + \sqrt{\mathbb{V}[\varepsilon(0)]}\Bigr)\biggr) \geq \mathtt{C}\frac{b}{2e}.
\end{align*}
\paragraph*{Proof of Equation~\eqref{sa-eq: tilde iota inverse}.} Let $F$ be the cummulative distribution function of $\mathbf{x}_i$. Suppose $1 \leq k \leq n/2$. Then $F(\mathbf{x}_{(k)}) \sim \mathsf{Beta}(k, M - k + 1)$. By a Bernstein bound for Beta variables \cite[Theorem 1]{skorski_2023}, we have for all $\epsilon > 0$,
\begin{align*}
\mathbb{P}(F(\mathbf{x}_{(k)}) > k/M - \epsilon) \leq \exp \Big( - \frac{\epsilon^2}{2 v}\Big),
\end{align*}
where for large enough $n$,
\begin{align*}
v & = \frac{k (M - k + 1)}{(M + 1)^2 (M + 2)} \leq 2\frac{k}{M^2}.
\end{align*}
Hence with probability at least $1 - s_n^{-1}$,
\begin{align*}
F(\mathbf{x}_{(k)}) \geq k/M - 2\frac{\sqrt{\log (s_n) k}}{M} .
\end{align*}
Condition on $\mathbf{X} = (\mathbf{x}_1, \cdots, \mathbf{x}_n)$, $\mathds{1}(\tilde{\mathbf{x}}_i \leq \mathbf{x}_{(k)})$'s are i.i.d $\mathsf{Bernoulli}(F(\mathbf{x}_{(k)}))$. Hence condition on $\mathbf{X}$ and $\hat{\iota}$, with probability at least $1 - s_n^{-1}$,
\begin{align*}
\tilde{\imath}/N = N^{-1}\sum_{i = 1}^N \mathds{1}(\tilde{\mathbf{x}}_i \geq \mathbf{x}_{(\hat{\iota})}) \geq F(\mathbf{x}_{(\hat{\iota})}) - 2 \sqrt{\frac{\log(s_n) F(\mathbf{x}_{(\hat{\iota})})}{N}}.
\end{align*}
It follows that on $s_n \leq \hat{\iota} \leq M - s_n$, using boundedness of $(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0)$,
\begin{align*}
\mathbb{E} \Bigl[(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0)\Big|\hat{\iota} \Bigr] & = \mathbb{E} \Bigl[(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0), \mathds{1}(\tilde{\iota} \geq \hat{\iota}/8)\Big|\hat{\iota} \Bigr] + \mathbb{E} \Bigl[(1 - \frac{1}{\tilde{\iota}})\mathds{1}(\tilde{\iota} > 0), \mathds{1}(\tilde{\iota} < \hat{\iota}/8)\Big|\hat{\iota} \Bigr] \\
& = 1 + O(\frac{8}{\hat{\iota}}) + O(s_n^{-1}) \\
& = 1 + O(s_n^{-1}).
\end{align*}
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency honest reg}}
For simplicity, denote $\hat\tau^{\mathtt{HON}}_{\mathtt{DIM}}$ by $\hat{\tau}$. Since given the partition $\mathcal{P}$ chosen by $\mathcal{D}_{\mathsf{T}}$, Equation~\eqref{sa-eq: dim least square representation} is still satisfied. We can use the same argument in the proof of Theorem~\ref{sa-thm: L2 consistency NSS reg} condition on $\mathcal{D}_{\mathsf{T}}$ to get
\begin{align*}
\mathbb{E}_{\mathcal{D}_{\tau}}[\lVert \hat{\tau} - \tau \rVert^2 | \mathcal{D}_{\mathsf{T}}]
\leq C \frac{2^K \log(n_{\tau})^5}{n_{\tau}},
\end{align*}
where $C$ is a positive constant that only depends on $\xi$, $\mu$ and the distribution of $\varepsilon_i(0), \varepsilon_i(1)$. In particular, the expectation is taken with respect to $\mathcal{D}_{\tau}$ with effective sample size $n_{\tau}$.
Since condition on $\mathcal{D}_{\mathsf{T}}$, the partition $\mathcal{P}$ is fixed, we can use the same argument as in Theorem~\ref{sa-thm: L2 consistency honest} to show that $\hat{g}$ lies in a class $\mathscr{H}_{n_{\mathsf{T}}}[\mathcal{P}]$ with covering number,
\begin{align*}
N(\varepsilon U, \mathscr{H}_{n_{\mathsf{T}}}[\mathcal{P}], \lVert \cdot \rVert_{P_{X^{n_{\tau}}}}) \leq \bigg(\frac{2}{\varepsilon}\bigg)^{2^K}, \qquad \varepsilon \in (0,1),
\end{align*}
when we assume $y_i(0)$ and $y_i(1)$ are bounded by $U$. In comparison, in the proof of Theorem~\ref{sa-thm: L2 consistency NSS reg}, we show $\hat{g}$ lies in $\mathscr{H}_{n_{\mathsf{T}}}$ with covering number
\begin{align*}
N(\varepsilon U, \mathscr{H}_{n_{\mathsf{T}}}, \lVert \cdot \rVert_{P_{X^{n_{\tau}}}}) \leq 2 (n_{\mathsf{T}} p)^{2^K} \bigg(\frac{417 e U^2}{\beta}\bigg)^{2^{K+1}}, \qquad \varepsilon \in (0,1).
\end{align*}
This improvement of covering number due to honesty means we can replace a $ \log(n_{\tau} p)$-penalty in the result of Theorem~\ref{sa-thm: L2 consistency NSS reg} by $\log(n_{\tau})$. Now uncondition over $\mathcal{D}_{\mathsf{T}}$ and using the fact that $\rho^{-1} \leq n_{\tau}/n_{\mathsf{T}} \leq \rho$, we get the conclusion.
\subsection{Proof of Theorem~\ref{thm:main-reg}}
The conclusion follows from Theorem~\ref{sa-thm: imbalance reg} and the same proof for Theorem~\ref{thm:main}.
\subsection{Proof of Theorem~\ref{sa-thm: L2 consistency reg}}
For simplicity, denote $\hat{\tau}^{\mathcal{X}}_{\mathtt{DIM}}(\mathbf{x}; K)$ by $\tilde \tau(T_K)$, and $N = n/(K+1)$ denotes the sample size for each folds in the $\mathtt{X}$ sample splitting scheme.
Let $\mathsf{t}_1, \mathsf{t}_2, \cdots, \mathsf{t}_{2^K}$ denote the $2^K$ leaf nodes in the decision tree, (if a node cannot be further refined, we duplicate the split indices and values at the next level). And let $N_1, N_2, \cdots, N_{2^K}$ and $m_1, m_2, \cdots, m_{2^K}$ denote the number of observations and the Lebesgue measure of the $2^K$ leaf nodes, respectively. Note that $\Vec{N} = (N_1, \cdots, N_{2^{K}})$ are independent of the $\tilde y_i $ data by the honest condition and the $x_i $ data per Assumption \ref{sa-ass:DGP}. As in the proof of Theorem~\ref{sa-thm: L2 consistency X}, we can show condition on $\Vec{N}$, $m_k \sim \mathsf{Beta}(N_k, N - N_k + 1)$
Thus, the IMSE can be bounded as follows: Since condition on $\Vec{N}$, $m_k$'s are independent to the refreshed samples $\tilde{d_i}, \tilde{\varepsilon}_i(0)$, $\tilde{\varepsilon}_i(1)$'s, we have
\begin{align*}
& \mathbb{E}\Bigg[\int_{\mathcal{X}}(\tilde \tau(T_K)(x) - \tau)^2 \mathbb{P}_x(dx)\Bigg] \\
& =
\sum_{k=1}^{2^K}\mathbb{E}\Bigg[m_k\Bigg(\frac{\sum_{\mathbf{x}_i \in \mathsf{t}_k} \tilde{d}_i \tilde{\varepsilon}_i(1)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} \tilde{d}_i} - \frac{\sum_{\mathbf{x}_i \in \mathsf{t}_k} (1 - \tilde{d}_i) \tilde{\varepsilon}_i(0)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} 1 - \tilde{d}_i}\Bigg)^2\Bigg]
\\ & \leq
\sum_{k=1}^{2^K}\mathbb{E}\Bigg[\mathbb{E} [m_k |\vec{N}] \; \mathbb{E} \bigg[\bigg(\frac{\sum_{\mathbf{x}_i \in \mathsf{t}_k} \tilde{d}_i \tilde{\varepsilon}_i(1)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} \tilde{d}_i} - \frac{\sum_{\mathbf{x}_i \in \mathsf{t}_k} (1 - \tilde{d}_i) \tilde{\varepsilon}_i(0)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} 1 - \tilde{d}_i}\bigg)^2\bigg|\vec{N}\bigg]\Bigg] \\
& \leq \sum_{k=1}^{2^K}\mathbb{E}\Bigg[\frac{N_k}{N} \bigg(\frac{\mathds{1}(\sum_{\mathbf{x}_i \in \mathsf{t}_k} d_i > 0)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} d_i} + \frac{\mathds{1}(\sum_{\mathbf{x}_i \in \mathsf{t}_k} 1 - d_i > 0)}{\sum_{\mathbf{x}_i \in \mathsf{t}_k} 1 - d_i}\bigg)\Bigg] \; \max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\}.
\end{align*}
Notice that condition on $\Vec{N}$, $\sum_{\mathbf{x}_i \in \mathsf{t}_k} d_i \sim \mathsf{Bin}(N_k, \xi)$ and $\sum_{\mathbf{x}_i \in \mathsf{t}_k} 1 - d_i \sim \mathsf{Bin}(N_k, 1 - \xi)$. Using the fact that for a binomial random variable $W \sim \mathsf{Bin}(n,p)$, we have
\begin{align*}
\mathbb{E}\bigg[\frac{1}{W} \mathds{1}(W > 0) \bigg] \leq \frac{C}{n p C_p},
\end{align*}
where $C$ is an absolute constant, and $C_p$ is some constant that only depends on $p$. It follows that
\begin{align*}
& \mathbb{E}\Bigg[\int_{\mathcal{X}}(\tilde \tau(T_K)(x) - \tau)^2 \mathbb{P}_x(dx)\Bigg] \\
& \leq \sum_{k=1}^{2^K}\mathbb{E}\Bigg[\frac{N_k}{N} \bigg(\frac{1}{N_k \xi} + \frac{1}{N_k (1 - \xi)}\bigg)\Bigg] \; \max\{\mathbb{V}[\varepsilon_i(0)], \mathbb{V}[\varepsilon_i(1)]\} \\
& \lesssim \frac{2^K}{N}.
\end{align*}
\subsection{Proof of Lemma~\ref{lem: approximation fit based -- balanced region}}
Assume w.l.o.g. $k \leq n/2$, since the case of $k > n/2$ can be dealt with by symmetry. From the proof of Lemma~\ref{lem: approximation -- balanced region},
\begin{align*}
\sup_{r_n \leq k < n - r_n} \frac{k (n - k)}{n} \bigg|(\hat \mu_{L,0}(k,\ell) - \hat \mu_{R,0}(k,\ell))^2 - (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2\bigg| = O_\mathbb{P} \bigg(\frac{\log \log n}{\sqrt{r_n}}\bigg).
\end{align*}
Moreover, the proof of the term $R_1$ in Lemma~\ref{lem: approximation -- balanced region} implies
\begin{align*}
\sup_{r_n \leq k < n - r_n} ((\hat \mu_{L,0}(k,\ell) - \hat \mu_{R,0}(k,\ell))^2 + (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2) = O_\mathbb{P} \bigg(\frac{\log \log n}{r_n}\bigg).
\end{align*}
Now we consider the randomness induced by $n_0, n_{L,0}, n_{R,0}$. By Theorem A.4.1 in \cite{csorgo1997limit},
\begin{align*}
\max_{r_n \leq k < n - r_n} \sqrt{k} \cdot \bigg|\frac{1}{k}\sum_{i = 1}^k (\frac{d_i}{\xi} - 1)\bigg|
= O_{\mathbb{P}} (\sqrt{\log \log n}),
\end{align*}
which implies
\begin{align*}
\sup_{r_n \leq k < n - r_n}\bigg|\frac{n_{L,0}(k) n_{R,0(k)}}{n_0} - (1 - \xi)\frac{k (n - k)}{n}\bigg| = O_\mathbb{P}(\sqrt{r_n \log \log n}).
\end{align*}
Putting together, triangle inequality implies
\begin{align*}
\max_{1 \leq \ell \leq p}\max_{r_n \leq k < n - r_n} \Big|\mathscr{I}^{\text{SSE}}(k,\ell) - \mathscr{I}^{\text{prox}}(k,\ell)\Big| = O_{\mathbb{P}} \bigg(\frac{\log \log (n)^{3/2}}{r_n^{1/2}}\bigg).
\end{align*}
\subsection{Proof of Lemma~\ref{lem: approximation fit based -- imbalanced region}}
The proof of Lemma~\ref{lem: approximation -- imbalanced region} implies that
\begin{align*}
\max_{1 \leq \ell \leq p} \max_{1 \leq k \leq s_n} k\bigg|(\hat \mu_{L,0}(k,\ell) - \hat \mu_{R,0}(k,\ell))^2 - (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2\bigg| = O_\mathbb{P}(\alpha_n),
\end{align*}
where $\alpha_n = \rho_n \log \log n + \frac{s_n}{n - s_n} \log \log n$, and \begin{align*}
\max_{1 \leq \ell \leq p} \max_{1 \leq k \leq s_n} k (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2 = O_\mathbb{P}(\rho_n \log \log n).
\end{align*}
Hence it also follows that
\begin{align*}
\max_{1 \leq \ell \leq p} \max_{1 \leq k \leq s_n} k (\hat \mu_{L,0}(k,\ell) - \hat \mu_{R,0}(k,\ell))^2 = O_\mathbb{P} \Big(\frac{s_n}{n - s_n} \log \log n + \alpha_n \Big) = O_\mathbb{P}(\alpha_n).
\end{align*}
When $1 \leq k \leq s_n$, we have $\frac{n_{L,0}(k) n_{R,0}(k)}{n_0} \leq n_{L,0}(k) \leq k$. The conclusion then follows.
\subsection{Proof of Theorem~\ref{thm: inconsistency fit}}
The proof is similar to the proof of Theorem~\ref{sa-thm: imbalance reg}, except that in Theorem~\ref{sa-thm: imbalance reg}, we approximate the split criterion by a time-transformed O-U process, while here we approximate the split criterion by the summation of \emph{two independent} time transformed O-U processes. We divide the proofs into two steps.
\begin{center}
\textbf{Step 1: Approximation of fit-based processes by ipw-based processes}
\end{center}
Let $0 < a < b < 1$. Let $\rho_n$ be a sequence of real numbers taking values in $(0,1)$ to be determined, and take $s_n = \exp((\log n)^{\rho_n})$. Then for large enough $n$, we have $s_n \leq n^a \leq n^b \leq n - s_n$. Consider the event
\begin{align*}
A_n = \{\exists \ell \in [p]: \max_{k \in [n]} \mathscr{I}^\text{SSE}(k,\ell) > \max_{k \notin [s_n,n - s_n]} \mathscr{I}^\text{SSE}(k,\ell)\}.
\end{align*}
Equation (A.4.18) and (A.4.20) imply that for each $\ell \in [p]$,
\begin{gather*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n} \mathscr{I}^{\text{prox}}(k,\ell) = O_\mathbb{P}(\rho_n \log \log(n)), \\
\max_{s_n \leq k \leq n - s_n}\mathscr{I}^{\text{prox}}(k,\ell) = 2 \log \log (n)(1 + o_\mathbb{P}(1)).
\end{gather*}
Hence
\begin{align*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n}\mathscr{I}^{\text{prox}}(k,\ell) = o_\mathbb{P} \bigg(\max_{s_n \leq k \leq n - s_n}\mathscr{I}^{\text{prox}}(k,\ell)\bigg), \qquad \ell \in [p],
\end{align*}
Approximations results from Lemma~\ref{lem: approximation fit based -- balanced region} (taking $r_n = s_n$) and Lemma~\ref{lem: approximation fit based -- imbalanced region}, using the same argument as \emph{step 1} in the proof of Theorem~\ref{sa-thm: imbalance reg}, with $\log \log \log \log(n)/\log \log(n) \ll \rho_n \ll 1$, then implies
\begin{align*}
\max_{1 \leq k \leq s_n, n - s_n \leq k \leq n}\mathscr{I}^{\text{SSE}}(k,\ell) = o_\mathbb{P} \bigg(\max_{s_n \leq k \leq n - s_n}\mathscr{I}^{\text{SSE}}(k,\ell)\bigg), \qquad \ell \in [p].
\end{align*}
Using a union bound, we get $\mathbb{P}(A_n) \rightarrow 0$ as $n \rightarrow \infty$. Observe that on the event $A_n^c$, the argmax for $\mathscr{I}^{\text{SSE}}$ should be inside $[s_n, n - s_n]$. Hence
\begin{align*}
& \mathbb{P}\Big(\exists \ell \in [p]: \max_k\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}^{\text{SSE}}(k,j), \; \max_{k}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{k \notin [n^a,n^b]}\mathscr{I}^{\text{SSE}}(k,\ell) \Big) \\
& \geq \mathbb{P}\Big(\exists \ell \in [p]: \max_k\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{k,j\neq \ell}\mathscr{I}^{\text{SSE}}(k,j), \; \max_{k}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{k \notin [n^a,n^b]}\mathscr{I}^{\text{SSE}}(k,\ell) \text{ and } A_n^c \Big) - \mathbb{P}(A_n) \\
& \geq \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j), \\
& \qquad \qquad \qquad \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,\ell) \Big) - 2 \mathbb{P}(A_n).
\end{align*}
Now we focus on the first term. By symmetry in the $p$ coordinates,
\begin{align*}
& \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,\ell) \Big) \\
& = p \mathbb{P}\Big(\max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{SSE}}(k,1) > \max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\text{SSE}}(k,1) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,1) \Big) \\
& \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j) < z, \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\text{SSE}}(k,1) > z > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,1) \Big) \\
& \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j) < z, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,1) < z\Big) \\
& \qquad \qquad - p \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j) < z, \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{SSE}}(k,1) < z\Big).
\end{align*}
Then using the fact that $\mathscr{I}^\text{prox}(k,\ell)$ approximates $\mathscr{I}^\text{SSE}(k,\ell)$ from Lemma~\ref{lem: approximation -- balanced region}, we have
\begin{align}\label{eq:prox-prob}
\nonumber & \mathbb{P}\Big(\exists \ell \in [p]: \max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{j\neq \ell \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,j), \max_{k \in [s_n,n-s_n]}\mathscr{I}^{\text{SSE}}(k,\ell) > \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{SSE}}(k,\ell) \Big) \\
\nonumber & \geq p \sup_{z \in \mathbb{R}} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{prox}}(k,j) < z - v_n, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}} \mathscr{I}^{\text{prox}}(k,1) < z - v_n\Big) \\
& \qquad \qquad - p \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{prox}}(k,j) < z + v_n, \max_{k \in [s_n, n - s_n]} \mathscr{I}^{\text{prox}}(k,1) < z + v_n\Big),
\end{align}
where $v_n = O_\mathbb{P}(\log \log (n) s_n^{-1/2})$.
\begin{center}
\textbf{Step 2: Gaussian approximation of IPW partial sums}
\end{center}
Recall that
\begin{align}\label{eq:prox}
\mathscr{I}^\text{prox}(k, \ell) = & (1 - \xi) \frac{k (n - k)}{n} (\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell))^2 + \xi \frac{k (n - k)}{n} (\bar \mu_{L,1}(k,\ell) - \bar \mu_{R,1}(k,\ell))^2.
\end{align}
and we will show that high dimensional random vector $\Xi$ from concatenating $(\sqrt{(1 - \xi) \frac{k (n - k)}{n}}(\bar \mu_{L,0}(k,\ell) - \bar \mu_{R,0}(k,\ell)): k \in [n], \ell \in [p])$ and $(\sqrt{(1 - \xi) \frac{k (n - k)}{n}}(\bar \mu_{L,1}(k,\ell) - \bar \mu_{R,1}(k,\ell)): k \in [n], \ell \in [p])$ can be approximated by a Gaussian random vector with the same covariance structure. The proof will still be based on writing $\Xi$ as $\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{C}_i$ where
\begin{align*}
\mathbf{C}_i = & \Big(\sqrt{n}\Big(\Big(\sqrt{\frac{n}{k(n-k)}} (\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n}): r_n \leq k \leq n - r_n\Big)^\top: 1 \leq \ell \leq p\Big)^\top \frac{1 - d_i}{1 - \xi}\varepsilon_i(0), \\
& \sqrt{n}\Big(\Big(\sqrt{\frac{n}{k(n-k)}} (\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n}): r_n \leq k \leq n - r_n\Big)^\top: 1 \leq \ell \leq p\Big)^\top \frac{d_i}{ \xi}\varepsilon_i(1)\Big)^\top,
\end{align*}
where $\# \pi^{\ell}$ denotes the inverse mapping of $\pi^{\ell}$, as in the proof of Theorem~\ref{sa-thm:master}.
Notice that the random vectors are $2 n p$ dimensional. For notational simplicity, in what follows, denote by $\mathbf{e}_{t,k,\ell}$ the indicator of the position corresponding to $\sqrt{(1 - \xi) \frac{k (n - k)}{n}}(\bar \mu_{L,t}(k,\ell) - \bar \mu_{R,t}(k,\ell))$, $t = 0,1$, $k \in [n]$, $\ell \in [p]$.
However, the format of Equation~\eqref{eq:prox} induces a different geometry when approximating probabilities in Equation~\eqref{eq:prox-prob}. Instead of high dimensional CLT for hyper-rectangles, we consider the class of simple convex sets \cite[Section 3.1]{chernozhukov2017central}.
Let $\mathcal{J}$ be a subset of $[n] \times [p]$. Consider the class of closed convex sets $\mathcal{A}$ containing sets of the form
\begin{align}\label{eq:clt-sets}
A = \{\mathbf{u} \in \mathbb{R}^{2 n p}: (\mathbf{e}_{0,k,\ell}^\top \mathbf{u}, \mathbf{e}_{1,k,\ell}^\top \mathbf{u}) \in B_2(s_{k,\ell}), s_{k,\ell} \in (0,n], (k, \ell) \in \mathcal{J}\},
\end{align}
where $B_2(r)$ denotes the Euclidean ball centered at $\mathbf{0}$ with radius $r$ in $\mathbb{R}^2$. That is, the class $\mathcal{A}$ contains intersections of cylinders $\{\mathbf{u} \in \mathbb{R}^{2np}: \lVert {(\mathbf{e}_{j_1}^\top \mathbf{u}, \mathbf{e}_{j_2}^\top \mathbf{u})} \rVert_2 \leq s\}$. Notice that for $z \in (0, n]$, the event in Equation~\eqref{eq:prox} (inside $\sup z$) can be characterized as the high dimensional vector $\Xi$ lies in a set in $\mathcal{A}$.
For each $A \in \mathcal{A}$, we consider its approximation by simple convex sets. For each $B_2(r)$, denote by $B_2^{\text{in},n}(r)$ and $B_2^{\text{out},n}(r)$ its inscribed and circumscribed regular $n^2$-gon. Take $m = n^2 |J|$. Then for each $A \in \mathcal{A}$ of the form \eqref{eq:clt-sets}, take
\begin{align*}
A^m = \{\mathbf{u} \in \mathbb{R}^{2 n p}: (\mathbf{e}_{0,k,\ell}^\top \mathbf{u}, \mathbf{e}_{1,k,\ell}^\top \mathbf{u}) \in B_2^{\text{in,n}}(s_{k,\ell}), s_{k,\ell} \in (0,n], (k, \ell) \in \mathcal{J}\},
\end{align*}
and
\begin{align*}
A^{m,\epsilon} = \{\mathbf{u} \in \mathbb{R}^{2 n p}: (\mathbf{e}_{0,k,\ell}^\top \mathbf{u}, \mathbf{e}_{1,k,\ell}^\top \mathbf{u}) \in B_2^{\text{out,n}}(s_{k,\ell}), s_{k,\ell} \in (0,n], (k, \ell) \in \mathcal{J}\}.
\end{align*}
Then $A^m \subseteq A \subseteq A^{m,\epsilon}$. Moreover, denote by $\mathcal{V}(A^m)$ the set consisting of $m$ unit vectors that are outward normal to the facets of $A^m$. Then $A^m$ can be alternatively characterized by
\begin{align*}
A^m = \cup_{\mathbf{v} \in \mathcal{V}(A^m)} \{\mathbf{w} \in \mathbb{R}^{2 n p}: \mathbf{w}^\top \mathbf{v} \leq S_A(\mathbf{v})\}, \qquad S_A(\mathbf{v}) = \sup \{\mathbf{w}^\top \mathbf{v}: \mathbf{w} \in A\}.
\end{align*}
Then we can analogously characterize $A^{m,\epsilon}$ by
\begin{align*}
A^{m,\epsilon} = \cup_{\mathbf{v} \in \mathcal{V}(A^m)} \{\mathbf{w} \in \mathbb{R}^{2 n p}: \mathbf{w}^\top \mathbf{v} \leq S_A(\mathbf{v}) + \epsilon_{\mathbf{v}}\}, \qquad S_A(\mathbf{v}) = \sup \{\mathbf{w}^\top \mathbf{v}: \mathbf{w} \in A\},
\end{align*}
where $\epsilon_{\mathbf{v}} \leq n^{-1}$ for large enough $n$. This shows our class $\mathcal{A}$ is a subclass of $\mathcal{A}^\text{si}(1, 3)$ (see \cite[Section 3.1]{chernozhukov2017central}). Now we check its conditions (M.1'), (M.2') and (E.1'). Let $\mathbf{v} \in \mathcal{V}(A^m)$. The definition of $A^m$ implies $\mathbf{v} = v_{0,k,\ell} \mathbf{e}_{0,k,\ell} + v_{1,k,\ell} \mathbf{e}_{1,k,\ell}$ for some $(k,\ell) \in \mathcal{J}$, and $v_{0,k,\ell}^2 + v_{1,k,\ell}^2 = 1$. Let $\mathbf{v} \in \mathcal{V}(A^m)$.
\begin{align*}
& \frac{1}{n}\sum_{i = 1}^n \mathbb{E}[|\mathbf{v}^\top \mathbf{C}_i|^2] \\
& = \frac{1}{n} \sum_{i = 1}^n \mathbb{E} \bigg[\bigg(v_{0,k,\ell} \frac{n}{\sqrt{k (n - k)}}(\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n})\frac{1 - d_i}{1 - \xi} \varepsilon_i(0) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad+ v_{1,k,\ell} \frac{n}{\sqrt{k (n - k)}}(\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n})\frac{d_i}{\xi} \varepsilon_i(1)\bigg)^2\bigg] \\
& = \frac{1}{n}\bigg(\frac{n}{\sqrt{k (n - k)}}\bigg)^2\sum_{i =1}^n \bigg\{v_{0,k,\ell}^2 \mathbb{E} \bigg[\bigg((\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n})\frac{1 - d_i}{1 - \xi} \varepsilon_i(0) \bigg)^2 \bigg] \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + v_{1,k,\ell}^2 \mathbb{E} \bigg[\bigg((\mathds{1}(\#\pi^{\ell}(i) \leq k) - \frac{k}{n})\frac{d_i}{\xi} \varepsilon_i(1) \bigg)^2 \bigg] \bigg\} \\
& \geq \min \{\mathbb{V}[(1 - \xi)^{-1}(1 - d_i)\varepsilon_i(0)], \mathbb{V}[\xi^{-1} d_i \varepsilon_i(1)]\},
\end{align*}
which verifies (M.1'). The fact that only two entries of $\mathbf{v}$ are nonzero and $v_{0,k,\ell}^2 + v_{1,k,\ell}^2 = 1$ implies that
\begin{align*}
n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{v}^\top \mathbf{C}_i|^3] \leq 4 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{0,k,\ell}^\top \mathbf{C}_i|^3] + 4 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{1,k,\ell}^\top \mathbf{C}_i|^3] \lesssim \sqrt{n / r_n},
\end{align*}
where the last inequality is from the calculation in Equation~\eqref{eq:third moment coupling}, and this verifies (M.2') for the third moment. Moreover,
\begin{align*}
n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{v}^\top \mathbf{C}_i|^4]
& \leq 8 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{0,k,\ell}^\top \mathbf{C}_i|^4] + 8 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{1,k,\ell}^\top \mathbf{C}_i|^4] \\
& \leq \sqrt{n / r_n} 8 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{0,k,\ell}^\top \mathbf{C}_i|^3] + \sqrt{n / r_n} 8 n^{-1}\sum_{i = 1}^n \mathbb{E}[|\mathbf{e}_{1,k,\ell}^\top \mathbf{C}_i|^3] \\
& \lesssim n / r_n.
\end{align*}
The same logic shows that $\mathbb{E}[\exp(|\mathbf{v}^\top \mathbf{C}_i|/(K\sqrt{n / r_n}))] \leq 2$, where $K$ is an absolute constant. Putting together, we verify conditions (M.2') and (E.1') with $B_n = \sqrt{n / r_n}$. Hence by \cite[Proposition 3.1]{chernozhukov2017central}, there exists mean-zero random vectors $\boldsymbol{D}_i \sim N(\mathbf{0}, \mathbb{E}[\mathbf{C}_i \mathbf{C}_i^\top])$ such that
\begin{align}\label{eq:fit-clt}
\sup_{A \in \mathcal{A}} |\mathbb{P}(n^{-1/2}\sum_{i = 1}^n \mathbf{C}_i \in A) - \mathbb{P}(n^{-1/2} \sum_{i =1}^n \boldsymbol{D}_i \in A)| \lesssim \bigg(\frac{\log^7(n)}{r_n}\bigg)^{1/6}.
\end{align}
\begin{center}
\textbf{Step 4: Gaussian-to-Gaussian Approximation}
\end{center}
Observe that for any $k_1, k_2 \in [n], \ell_1, \ell_2 \in [p]$, we have $\operatorname{Cov}[\mathbf{e}_{0,k_1,\ell_1}^\top \mathbf{C}_i, \mathbf{e}_{1, k_2, \ell_2}^\top \mathbf{C}_i] = 0$. The same calculation as \emph{Multivariate Case Step 2} for the proof of Theorem~\ref{sa-thm:master} implies we can replace $\boldsymbol{D}_i$ by another mean-zero Gaussian random vector $\mathbf{Z}_i$ such that
\begin{align*}
\operatorname{Cov}[\mathbf{e}_{t_1, k_1, \ell_1}^\top \mathbf{Z}_i, \mathbf{e}_{t_2, k_2, \ell_2}^\top \mathbf{Z}_i] & = \begin{cases}
\operatorname{Cov}[\mathbf{e}_{t_1, k_1, \ell_1}^\top \boldsymbol{D}_i, \mathbf{e}_{t_2, k_2, \ell_2}^\top \boldsymbol{D}_i], & \text{ if } \ell_1 = \ell_2, \\
0, & \text{ otherwise}.
\end{cases}
\end{align*}
We want to show $\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i$ is close to $\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i$, measured by the probability of taking value in sets from $\mathcal{A}$ defined at Equation~\eqref{eq:clt-sets}. We omit details for simplicity, but illustrate the main skeleton here. As in Step 2, Nazarov inequality implies we only need to work on the $m$-generated convex approximation with $\epsilon$ precision $A^m = A^m(A)$ for $A \in \mathcal{A}$, for a reason given in \cite[proof of Proposition 3.1]{chernozhukov2017central}. Moreover, $\mathbb{P}(\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i \in A^m) = \mathbb{P}((\mathbf{v}^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i ))_{\mathbf{v} \in \mathcal{V}(A^m)} \leq \mathbf{t})$ for some $\mathbf{t} \in \mathbb{R}^m$. Hence we only need to show
\begin{align*}
\sup_{\mathbf{t} \in \mathbb{R}^m} \bigg|\mathbb{P}((\mathbf{v}^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i ))_{\mathbf{v} \in \mathcal{V}(A^m)} \leq \mathbf{t}) - \mathbb{P}((\mathbf{v}^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i ))_{\mathbf{v} \in \mathcal{V}(A^m)} \leq \mathbf{t})
\bigg| = o(1).
\end{align*}
But the definition of $\mathcal{A}$ in Equation~\eqref{eq:clt-sets} implies for any $A \in \mathcal{A}, \mathbf{v} \in \mathcal{V}(A^m)$, there exists $\mathbf{e}_k, \mathbf{e}_j$ and $v_k^2 + v_j^2 = 1$ such that $\mathbf{v} = v_k \mathbf{e}_k + v_j \mathbf{e}_j$, with
\begin{align*}
\operatorname{Cov} \bigg[\mathbf{e}_k^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i ), \mathbf{e}_j^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i )\bigg] = \operatorname{Cov} \bigg[\mathbf{e}_k^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i ), \mathbf{e}_j^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i )\bigg] = 0,
\end{align*}
and hence
\begin{align*}
\min_{\mathbf{z} \in \mathcal{V}(A^m)} \mathbb{V} \bigg[\mathbf{v}^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i ))_{\mathbf{v} \in \mathcal{V}(A^m)}\bigg] \gtrsim 1.
\end{align*}
Together with Equation~\eqref{eq: matrix comparison}, we know
\begin{align*}
\max_{\mathbf{v}_1, \mathbf{v}_2 \in \mathcal{V}(A^m)} \bigg| \operatorname{Cov} \bigg[\mathbf{v}_1^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i ), \mathbf{v}_2^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \mathbf{Z}_i )\bigg] - \operatorname{Cov} \bigg[\mathbf{v}_1^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i ), \mathbf{v}_2^\top (\frac{1}{\sqrt{n}}\sum_{i = 1}^n \boldsymbol{D}_i )\bigg] \bigg| = O(r_n^{-1/2}).
\end{align*}
The Gaussian-to-Gaussian Comparison result \cite[Proposition 2.1]{chernozhuokov2022improved} them implies
\begin{align}\label{eq:fit-gaussian-comparison}
\sup_{A \in \mathcal{A}} |\mathbb{P}(n^{-1/2}\sum_{i = 1}^n \mathbf{Z}_i \in A) - \mathbb{P}(n^{-1/2} \sum_{i =1}^n \boldsymbol{D}_i \in A)| = O(\log(n) r_n^{-1/2}).
\end{align}
\begin{center}
\textbf{Step 5: Orstein-Uhlenbeck Process Calculations}
\end{center}
Now we revisit Equation~\eqref{eq:prox-prob}. Consider
\begin{align*}
\mathscr{I}^\text{Gauss}(k, \ell) = & (1 - \xi) \frac{k (n - k)}{n} (\tilde \mu_{L,0}(k,\ell) - \tilde \mu_{R,0}(k,\ell))^2 + \xi \frac{k (n - k)}{n} (\tilde \mu_{L,1}(k,\ell) - \tilde \mu_{R,1}(k,\ell))^2,
\end{align*}
with
\begin{align*}
& \tilde \mu_{L,0}(k,\ell) = \frac{1}{k}\sum_{i \leq k} u_{\pi_{\ell}(i)}, && \tilde \mu_{L,1}(k,\ell) = \frac{1}{k}\sum_{i \leq k} v_{\pi_{\ell}(i)}, \\
& \tilde \mu_{R,0}(k,\ell) = \frac{1}{n - k}\sum_{i > k} u_{\pi_{\ell}(i)}, && \tilde \mu_{R,1}(k,\ell) = \frac{1}{n - k}\sum_{i > k} v_{\pi_{\ell}(i)}.
\end{align*}
Equations~\eqref{eq:fit-clt} and \eqref{eq:fit-gaussian-comparison} imply that
\begin{align*}
& \sup_{z \in \in [-n,n]} \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{prox}}(k,j) < z - v_n, \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}} \mathscr{I}^{\text{prox}}(k,1) < z - v_n\Big) \\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{prox}}(k,j) < z + v_n, \max_{k \in [s_n, n - s_n]} \mathscr{I}^{\text{prox}}(k,1) < z + v_n\Big) \\
& = \sup_{z \in [-n, n]}\mathbb{P}\Big(\max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{Gauss}}(k,1) < z - v_n \Big)^{p-1} \mathbb{P} \Big( \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}} \mathscr{I}^{\text{Gauss}}(k,1) < z - v_n\Big) \\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{Gauss}}(k,j) < z + v_n \Big)^{p-1} \mathbb{P} \Big(\max_{k \in [s_n, n - s_n]} \mathscr{I}^{\text{Gauss}}(k,1) < z + v_n\Big) + o(1).
\end{align*}
The same argument as \cite[(A.4.25) to (A.4.37)]{csorgo1997limit} shows that there exists two independent standard Brownian bridges over $[0,1]$, $B_{n,L}$ and $B_{n,R}$, for each $n$, such that
\begin{gather*}
\bigg|\max_{k \in [s_n, n - s_n]} \sqrt{\mathscr{I}^\text{Gauss}(k,1)} - \sup_{t \in [s_n/n, 1 - s_n/n]} \sigma\sqrt{\frac{B_{n,L}^2}{t (1 - t)} + \frac{B_{n,R}^2}{t (1 - t)}}\bigg| = \epsilon_n, \\
\bigg|\max_{k \in [s_n, n - s_n] \setminus [n^a, n^b]} \sqrt{\mathscr{I}^\text{Gauss}(k,1)} - \sup_{t \in [s_n/n, 1 - s_n/n] \setminus [n^{1-a}, n^{1-b}]} \sigma\sqrt{\frac{B_{n,L}^2}{t (1 - t)} + \frac{B_{n,R}^2}{t (1 - t)}}\bigg| = \epsilon_n,
\end{gather*}
with $\sigma^2 = \mathbb{V}[\varepsilon_i(0)] = \mathbb{V}[\varepsilon_i(1)]$ and $\epsilon_n = o_{\mathbb{P}}((\log \log n)^{-1/2})$. Let $\{U_L(t): t \in \mathbb{R} \}$ and $\{U_R(t): t \in \mathbb{R}\}$ be two independent O-U processes with $\mathbb{E}[U_j(t)] = 0$ and $\mathbb{E}[U_j(s) U_j(t)] = e^{-|s-t|}$, $j = L,R$. Then
\begin{align*}
\bigg\{\bigg(\frac{B_{n,L}}{\sqrt{t(1 - t)}}, \frac{B_{n,R}}{\sqrt{t (1 - t)}}\bigg): t \in [0,1]\bigg\} \stackrel{d}{=} \{(U_L(\log(t/(1-t))), U_R(\log(t/(1 - t)))): t \in [0,1]\}.
\end{align*}
Take $N(t) = \lVert (U_L(t), U_R(t)) \rVert_2, t \in \mathbb{R}$. Then a time change and stationarity of O-U process implies
\begin{align*}
& \mathbb{P} \bigg(\sup_{t \in [1/n, 1 - 1/n] \setminus [n^{1-a}, n^{1-b}]} \sqrt{\frac{B_{n,L}^2}{t (1 - t)} + \frac{B_{n,R}^2}{t (1 - t)}} \leq y \bigg) \\
& = \mathbb{P} \bigg(\sup_{-\log(n - 1) \leq t < \log(n^{a-1}/(1 - n^{a-1}), \log(n^{b-1}/(1-n^{b-1})) < t \leq \log(n-1)} |N(t)| \leq y \bigg) \\
& = \mathbb{P} \bigg(\sup_{0 \leq t < \log(n^{a-1}(n - 1)/(1 - n^{a-1}), \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |N(t)| \leq y \bigg),
\end{align*}
and
\begin{align*}
\mathbb{P} \bigg(\sup_{t \in [1/n, 1 - 1/n] \setminus [n^{1-a}, n^{1-b}]} \sqrt{\frac{B_{n,L}^2}{t (1 - t)} + \frac{B_{n,R}^2}{t (1 - t)}} \leq y \bigg)
= \mathbb{P} \bigg(\sup_{0 \leq t < 2\log(n-1)} |N(t)| \leq y \bigg).
\end{align*}
An expansion based on \cite[Lemma 2.1]{horvath1993maximum} (Lemma TODO) gives for any $z \in \mathbb{R}$,
\begin{align*}
\mathbb{P} \bigg(\sup_{0 \leq t < c \log (n)} |N(t)| \leq \frac{z + 2 \log \log(n) + \log \log \log(n)}{\sqrt{2 \log \log (n)}} + \epsilon_n \bigg) = \exp(- e^{-z + \log(c)}) + o(1).
\end{align*}
Moreover, Gaussian correlation inequality \citep[Remark 3 (i)]{latala2017royen} and stationarity of O-U process implies
\begin{align*}
& \mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}}), \; \log\frac{n^{b-1}(n-1)}{1-n^{b-1}}) < t \leq 2\log(n-1)} |N(t)| < \frac{z + 2 \log \log(n) + \log \log \log(n)}{\sqrt{2 \log \log (n)}} + \epsilon_n\bigg) \\
& \geq
\mathbb{P}\bigg(\sup_{ 0 \leq t < \log(\frac{n^{a-1}(n-1)}{1-n^{a-1}})} |N(t)| < \frac{z + 2 \log \log(n) + \log \log \log(n)}{\sqrt{2 \log \log (n)}} + \epsilon_n \bigg)\\
& \qquad \qquad \cdot \mathbb{P}\bigg(\sup_{0 < t \leq \log(n^{1-b}(n-1)(1-n^{b-1}))} |N(t)| < \frac{z + 2 \log \log(n) + \log \log \log(n)}{\sqrt{2 \log \log (n)}} + \epsilon_n \bigg) \\
& = \exp(- 2 e^{-z + \log(2 - (b - a))}) + o(1).
\end{align*}
Putting together and choosing $z^\ast$ that maximizes $z \mapsto \exp(- 2 e^{-z + \log(2 - (b - a))}) - \exp(-2 e^{-z + \log(c)})$, we can get
\begin{align*}
& \sup_{z \in [-n, n]}\mathbb{P}\Big(\max_{k \in [s_n, n - s_n]}\mathscr{I}^{\text{Gauss}}(k,1) < z - v_n \Big)^{p-1} \mathbb{P} \Big( \max_{\substack{k \notin [n^a,n^b] \\ k \in [s_n, n - s_n]}} \mathscr{I}^{\text{Gauss}}(k,1) < z - v_n\Big) \\
& \qquad \qquad - \mathbb{P}\Big(\max_{\substack{j\neq 1 \\ k \in [s_n, n - s_n]}}\mathscr{I}^{\text{Gauss}}(k,j) < z + v_n \Big)^{p-1} \mathbb{P} \Big(\max_{k \in [s_n, n - s_n]} \mathscr{I}^{\text{Gauss}}(k,1) < z + v_n\Big) \\
& \geq \sup_z \exp\Big(-2(p-1)e^{-(z-\log(2))}\Big)\Big( \exp\Big(-2e^{-(z-\log(2-(b-a)))}\Big) - \exp\Big(-2e^{-(z-\log(2))}\Big)\Big) \\
& = \frac{b-a}{2p}\bigg(1-\frac{b-a}{2p}\bigg)^{\frac{2p}{b-a}-1} \\ & \geq \frac{b-a}{2pe}.
\end{align*}
Symmetry then implies for any $0 < a < b < 1$ and $\ell \in [p]$, we have
\begin{equation*}
\liminf_{n\to\infty} \mathbb{P}\big( n^{a} \leq \hat\imath_\text{SSE} \leq n^{b}, \hat \jmath_\text{SSE} = \ell \big) = \liminf_{n\to\infty} \mathbb{P}\big( n-n^{b} \leq \hat\imath_\text{SSE} \leq n-n^{a}, \hat \jmath_\text{SSE} = \ell \big) \geq \frac{b-a}{2pe}.
\end{equation*}
\subsection{Proof of Corollary~\ref{sa-coro:rates_reg_fit}}
Notice that although the splitting criteria is different from the regression tree, once cells are given the estimator given by the fit-based tree is exactly the same as the regression tree (see Section~\ref{sa-sec: reg causal}). Hence result can be proved based on Theorem~\ref{thm: inconsistency fit} and the same logic as Theorem~\ref{sa-thm:rates_reg}.
\subsection{Proof of Corollary~\ref{sa-coro: uniform minimax rates regression fit}}
Notice that although the splitting criteria is different from the regression tree, once cells are given the estimator given by the fit-based tree is exactly the same as the regression tree (see Section~\ref{sa-sec: reg causal}). Hence result can be proved based on Theorem~\ref{thm: inconsistency fit} and the same logic as Theorem~\ref{sa-thm: uniform minimax rates regression}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency NSS fit}}
Since the tree is constructed by minimizing the objective Equation~\eqref{eq: sse} iteratively. The empirical risk minimization property Equation~\eqref{sa-eq: empirical risk minimization} still holds. Hence the result follows from the same argument as the proof of Theorem~\ref{sa-thm: L2 consistency NSS reg}.
\subsection{Proof of Corollary~\ref{sa-coro: honest output reg fit}}
Notice that although the splitting criteria is different from the regression tree, once cells are given the estimator given by the fit-based tree is exactly the same as the regression tree (see Section~\ref{sa-sec: reg causal}). Hence result can be proved based on Theorem~\ref{thm: inconsistency fit} and the same logic as Theorem~\ref{sa-coro: honest output reg}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency honest fit}}
Since the tree is constructed by minimizing the objective Equation~\eqref{eq: sse} iteratively. The empirical risk minimization property Equation~\eqref{sa-eq: empirical risk minimization} still holds. Hence the result follows from the same argument as the proof of Theorem~\ref{sa-thm: L2 consistency honest reg}.
\subsection{Proof of Corollary~\ref{sa-coro:main-reg fit}}
Notice that although the splitting criteria is different from the regression tree, once cells are given the estimator given by the fit-based tree is exactly the same as the regression tree (see Section~\ref{sa-sec: reg causal}). Hence result can be proved based on Theorem~\ref{thm: inconsistency fit} and the same logic as Theorem~\ref{thm:main-reg}.
\subsection{Proof of Corollary~\ref{sa-coro: L2 consistency fit}}
The result follows from the same argument as Theorem~\ref{sa-thm: L2 consistency reg}.
\subsection{Proof of Lemma~\ref{sa-lem: unbiased}}
First, we consider $\mathtt{X}$ under Assumption~\ref{sa-assump: dgp-causal}. Since $(y_i, d_i)$'s are from dataset $\mathscr{D}_{\tau}$ independent to the dataset $\mathscr{D}_{\mathsf{T}_1}$ to $\mathscr{D}_{\mathsf{T}_K}$ for tree construction, it is easy to check that
\begin{align*}
\mathbb{E}[\hat{\tau}^{\mathcal{X}}_l(\mathbf{x};K)] = \mathbb{E}[\mathbb{E}[\hat{\tau}^{\mathcal{X}}_l(\mathbf{x};K)|\mathsf{T}, (\mathbf{x}_i)_{i \in \mathcal{D}_{\tau}}]] = \tau, \qquad l \in \{\mathtt{DIM}, \mathtt{IPW}, \mathtt{SSE}\}.
\end{align*}
Next, we consider $\mathtt{HON}$ under Assumption~\ref{sa-assump: dgp-causal}. Denote by $\mathsf{t}(\mathbf{x})$ the node that contains $\mathbf{x}$, and denote by $n(\mathsf{t})$ the \emph{local sample size} in $\mathcal{D}_{\tau}$, where $n(\mathsf{t}) = \sum_{i \in \mathcal{D}_{\tau}}\mathds{1}(\mathbf{x}_i \in \mathsf{t})$. Then
\begin{align*}
\mathbb{E}[\hat{\tau}_{\mathtt{IPW}}^{\mathtt{HON}}(\mathbf{x};K)|\mathcal{D}_{\mathsf{T}}]
& = \mathbb{E}[\hat{\tau}_{\mathtt{IPW}}^{\mathtt{HON}}(\mathbf{x};K) \mathds{1}(n(\mathsf{t}(\mathbf{x})) > 0)|\mathcal{D}_{\mathsf{T}}] + 0 \cdot \mathbb{P}(n(\mathsf{t}(\mathbf{x})) = 0 |\mathcal{D}_{\mathsf{T}}) \\
& = \mathbb{E}[\hat{\tau}_{\mathtt{IPW}}^{\mathtt{HON}}(\mathbf{x};K) |\mathcal{D}_{\mathsf{T}}, n(\mathsf{t}(\mathbf{x})) > 0] \mathbb{P}(n(\mathsf{t}(\mathbf{x})) > 0 |\mathcal{D}_{\mathsf{T}}) \\
& = \tau \mathbb{P}(n(\mathsf{t}(\mathbf{x})) > 0 |\mathcal{D}_{\mathsf{T}}),
\end{align*}
where in the third line, we have used the fact that $\varepsilon_i(0)$ and $\varepsilon_i(1)$ in $\mathcal{D}_{\tau}$ are independent to $\mathbf{x}_i$'s in $\mathcal{D}_{\tau}$ and the whole dataset $\mathcal{D}_{\mathsf{T}}$, with $\mathbb{E}[\varepsilon_i(0)] = \mathbb{E}[\varepsilon_i(1)] = 0$. Unconditioning over $\mathcal{D}_{\mathsf{T}}$, then we get
\begin{align*}
\mathbb{E}[\hat{\tau}_{\mathtt{IPW}}^{\mathtt{HON}}(\mathbf{x};K)] = \tau \mathbb{P}(n(\mathsf{t}(\mathbf{x})) > 0).
\end{align*}
The results for $\mathtt{DIM}$ and $\mathtt{SSE}$ can be obtained by similar arguments.
Finally, we consider $\mathtt{NSS}$ under Assumption~\ref{sa-assump: dgp-causal} and the additional symmetric error $\varepsilon_i(0)$, and $\varepsilon_i(1)$ assumption. We will use an induction assumption.
\textit{Base case: $K = 1$.} Due to the assumption that $\mu_0$ and $\mu_1$ are constant, we can rewrite the splitting criteria from Definition 2 in the main paper as
\begin{align}\label{sa-eq: split dim}
\nonumber \text{DIM} : \qquad \frac{n(\mathsf{t}_{\mathtt{L}})n(\mathsf{t}_{\mathtt{R}})}{n(\mathsf{t})}
\Big(& \frac{1}{n_{1}(\mathsf{t}_L)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_L} d_i \varepsilon_i(1)
- \frac{1}{n_{0}(\mathsf{t}_L)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_L} (1-d_i) \varepsilon_i(0)
\\
& - \frac{1}{n_{1}(\mathsf{t}_R)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_R} d_i \varepsilon_i(1)
+ \frac{1}{n_{0}(\mathsf{t}_R)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_R} (1-d_i) \varepsilon_i(0))\Big)^2,
\end{align}
and
\begin{align}\label{sa-eq: split ipw}
& \mathtt{IPW}: \qquad \frac{n(\mathsf{t}_{\mathtt{L}})n(\mathsf{t}_{\mathtt{R}})}{n(\mathsf{t}_L)}
\Big(\frac{1}{n(\mathsf{t}_L)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_L} (\frac{d_i}{\xi}\varepsilon_i(1) - \frac{1 - d_i}{1 - \xi} \varepsilon_i(0))
- \frac{1}{n(\mathsf{t}_R)} \sum_{i:\mathbf{x}_i \in \mathsf{t}_R} (\frac{d_i}{\xi}\varepsilon_i(1) - \frac{1 - d_i}{1 - \xi} \varepsilon_i(0)) \Big)^2,
\end{align}
and
\begin{align}\label{sa-eq: split sse}
\nonumber & \text{SSE}: \qquad \frac{n_1(\mathsf{t}_{\mathtt{L}})n_1(\mathsf{t}_{\mathtt{R}})}{n_1(\mathsf{t})}
\Big( \frac{1}{n_1(\mathsf{t}_{\mathtt{L}})} \sum_{i: \mathbf{x}_i \in \mathsf{t}_{\mathtt{L}}} d_i \varepsilon_i(1)
- \frac{1}{n_1(\mathsf{t}_{\mathtt{R}})} \sum_{i: \mathbf{x}_i \in \mathsf{t}_{\mathtt{R}}} d_i \varepsilon_i(1) \Big)^2 \\
& \qquad \qquad \qquad +
\frac{n_0(\mathsf{t}_{\mathtt{L}})n_0(\mathsf{t}_{\mathtt{R}})}{n_0(\mathsf{t})}
\Big( \frac{1}{n_0(\mathsf{t}_{\mathtt{L}})} \sum_{i: \mathbf{x}_i \in \mathsf{t}_{\mathtt{L}}} (1 - d_i) \varepsilon_i(0)
- \frac{1}{n_0(\mathsf{t}_{\mathtt{R}})} \sum_{i: \mathbf{x}_i \in \mathsf{t}_{\mathtt{R}}} (1 - d_i) \varepsilon_i(0) \Big)^2.
\end{align}
Denote the vector $\boldsymbol{\varepsilon} = (\varepsilon_1(0), \varepsilon_1(1), \cdots, \varepsilon_n(0), \varepsilon_n(1))$. Notice that for all three criteria, for any $\mathbf{d} = (d_1, \cdots, d_n)$ and $\mathsf{t}_L, \mathsf{t}_R$, $\varepsilon = \mathbf{u}$ and $\varepsilon = - \mathbf{u}$ give the same value. Hence condition on $\mathbf{d}$ and the data-driven split region $\hat{\mathsf{t}}_L$ and $\hat{\mathsf{t}}_R$, $\boldsymbol{\varepsilon}$ is symmetrically distributed around zero. It then follows from the form of the three estimators that all of them are unbiased.
\textit{Induction step: $K \geq 2$.} Each leaf node $\mathsf{t}$ in layer $K-1$ is further partitioned into $\mathsf{t}_L$ and $\mathsf{t}_R$ such that Equations~\eqref{sa-eq: split dim}, \eqref{sa-eq: split ipw} and \eqref{sa-eq: split sse} are maximized. The induction hypothesis is that condition on all leaf $\mathsf{t}$ in the $K-1$ th layer and $\mathbf{d}$, $\boldsymbol{\varepsilon}$ is symmetrically distributed around zero. Again for all three criteria, given $K-1$th leaf node $\mathsf{t}$, for any $\mathbf{d} = (d_1, \cdots, d_n)$ and $\mathsf{t}_L, \mathsf{t}_R$, $\varepsilon = \mathbf{u}$ and $\varepsilon = - \mathbf{u}$ give the same value. Hence the resulting $K$th level $\hat{\mathsf{t}}_L$ and $\hat{\mathsf{t}}_R$ are such that condition on $\mathbf{d}$ and the data-driven split region $\hat{\mathsf{t}}_L$ and $\hat{\mathsf{t}}_R$, $\boldsymbol{\varepsilon}$ is symmetrically distributed around zero, making the estimators unbiased.
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