Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

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Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps


\maketitle
\begin{abstract}
Volatility estimation is a central problem in financial econometrics, but becomes particularly challenging when jump activity is high, a phenomenon observed empirically in highly traded financial securities. In this paper, we revisit the problem of spot volatility estimation for an Itô semimartingale with jumps of unbounded variation. We construct truncated kernel-based estimators and debiased variants that extend the efficiency frontier for spot volatility estimation in terms of the jump activity index $Y$, raising the previous bound $Y<4/3$ to $Y<20/11$, thereby covering nearly the entire admissible range $Y<2$.  Compared with earlier work, our approach attains smaller asymptotic variances through the use of unbounded kernels, is simpler to implement, and has broader applicability under more flexible model assumptions. A comprehensive simulation study confirms that our procedures substantially outperform competing methods in finite samples.
\end{abstract}
\section{Introduction}


 Spot volatility estimation is a central problem in financial econometrics, underpinning numerous critical applications ranging from short-horizon risk management to derivatives pricing and hedging. Formally, the core task is to estimate the coefficient $\sigma_\tau$ of the Brownian component of an It\^o semimartingale $X$ at a fixed point of time $\tau$ given high-frequency observations. For recent developments, we refer to \cite{LIU2018,figueroa2020optimal,Figueroa-Lopez_Wu_2024,mancino2022asymptotic}, as well as the monographs \cite{jacod2012discretization,ait2014high} for in-depth treatments.



Widely observed empirically in high-frequency financial data, jumps are a salient feature of price dynamics, separate from fluctuations captured by diffusive movements in $X$, and must be carefully accounted for when estimating volatility and related functionals of $X$.  When jump activity is high, however, standard estimation methods face inherent limitations, as estimation difficulty increases with the degree of jump activity of the underlying process. Three qualitatively distinct regimes can be distinguished: jumps of finite activity,  infinite activity jumps of bounded variation, and infinite activity jumps of unbounded variation, the latter posing the greatest challenge, and the setting of the present work.

A fundamental approach to handling jumps is through truncation, introduced by \cite{Mancini2004}.  Establishing jump-robustness for truncation-based methods classically amounts to two steps: suppose $\hat\theta_n(\Delta_1^nX,\dots,\Delta^n_n X)$ is an estimator of some functional $\theta$ of the continuous component $X^c$ of $X$, where $\Delta_i^nU:=U_{t_{i}}-U_{t_{i-1}}$ denotes the $i^{th}$ increment of a generic process $U$ sampled at times $0<t_0<\dots<t_n=T$ during a finite time period $[0,T]$.  In the first step, one shows that the (infeasible) estimator $\hat\theta^{\text{Cont}}_n:=\hat{\theta}_n(\Delta_1^nX^c,\dots,\Delta_n^n X^c)$ is consistent and satisfies a central limit theorem (CLT) with, say, rate $r_n$:
\begin{equation}\label{M1aTCLT}
    \frac{1}{r_n}\left(\hat{\theta}_n(\Delta_1^nX^c,\dots,\Delta_n^n X^c)-\theta\right)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,V_\theta).
\end{equation}
Second, it is shown that, by choosing an appropriate threshold level $v_n\searrow{}0$, the truncated or thresholded version $\hat\theta^{Trunc}_n:=\hat{\theta}_n(\Delta_1^nX\mathbf{1}_{\{|\Delta_1^n X|\leq  v_n\}},\dots,\Delta_n X^n\mathbf{1}_{\{|\Delta_n^n X|\leq  v_n\}})$ is close enough to $\hat\theta^{Cont}$ so that
\begin{equation}\label{M1bTCLT}
    \frac{1}{r_n}\left(\hat\theta_n^{\text{Trunc}}-\hat\theta_n^{\text{Cont}}\right)=o_P(1).
\end{equation}
An emblematic example of this approach arises in the estimation of the integrated variance or volatility $IV:=\int_0^T \sigma^2_sds$, where one passes from the quantity
\begin{equation}
    \widehat{IV}^{\text{Cont}}:=\sum_{i=1}^n(\Delta_{i}^nX^{c})^2,
\end{equation}
to its truncated version
\begin{equation}\label{TRVEst}
 \widehat{IV}^{\text{Trunc}}:=\sum_{i=1}^n(\Delta_{i}^nX)^2{\bf 1}_{\{|\Delta_i^nX|\leq v_n\}}.
\end{equation}
By (essentially) employing the steps \eqref{M1aTCLT}-\eqref{M1bTCLT}, it has been established that in the bounded variation case  \cite{mancini2009non,cont2011nonparametric}, an efficient CLT (see, e.g., \cite{jacod:reiss:2014}) is possible:
\[
    \sqrt{n}\left(\widehat{IV}^{\text{Trunc}}-IV\right)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(0,2\int_0^T\sigma_s^4ds\right).
\]
However, with unbounded-variation jumps, this approach alone falls short.  It was proved by \cite{cont2011nonparametric,mancini:2011} that $\sqrt{n}\big(\widehat{IV}^{\text{Trunc}}-IV\big)\stackrel{P}{\longrightarrow}\infty$ in the presence of an infinite variation symmetric stable L\'evy process, precisely because the bias of $\widehat{IV}^{\text{Trunc}}$ -- and thus the difference $(\widehat{IV}^{\text{Trunc}} - \widehat{IV}^{\text{Cont}})$ -- vanishes too slowly.


Two methodological advances have since moved the field forward. The first is due to Jacod and Todorov \cite{Jacod2014,jacod:todorov:2016}, who proposed an approach for integrated variance estimation based on the empirical characteristic function of the rescaled increments $\Delta_i^nX/\sqrt{\Delta_n}$, where $\Delta_n$ is the time span of the increment. After applying a suitable logarithmic transformation and debiasing procedure, this approach admits a CLT with optimal rate and variance, even in the case of unbounded variation jumps.  However, with this method, full (rate and variance) efficiency is attainable only under specific conditions that require either the jump activity index $Y$ to satisfy $Y<3/2$, or under additional restrictive jump symmetry assumptions.  The second major advance was introduced more recently by    \cite{BONIECE2024}, who showed that a debiasing method similar to \cite{Jacod2014} applied to the realized truncated variations \eqref{TRVEst} leads to a fully efficient estimator for any $Y\leq 8/5$ without any symmetry conditions.




A natural next question is how these insights extend to the problem of spot volatility estimation. A standard and widely used approach to constructing spot volatility estimators is to apply kernel smoothing to an estimator of integrated variance,
\[
    \hat\sigma^2_\tau=\int K_b(t-\tau)d \widehat{IV}_t,
\]
where $\widehat{IV}_t$ denotes an estimator of $IV_t=\int_0^t \sigma_s^2ds$ and $K_b(u):=K(u/b)/b$ for a chosen kernel $K$ and a bandwidth $b>0$.
For instance, using the realized quadratic variation $\widehat{IV}_t=\sum_{i=1}^{[nt]}(\Delta_i ^nX)^2$, this reduces to
\begin{equation}\label{KrnlSpotVol0}
    \hat\sigma^2_\tau=\sum_{i=1}^n K_b(t_{i-1}-\tau)(\Delta_i^n X)^2,
\end{equation}
as proposed in \cite{kristensen2010nonparametric,fan2008spot}.

Spot volatility estimators differ markedly from their integrated volatility counterparts, with their local nature altering the efficiency picture in subtle but important ways: on the one hand, optimal convergence rates are of order $n^{-1/4}$ (compared with $n^{-1/2}$ in the integrated volatility case); on the other hand,  kernel-based estimators enjoy a degree of built-in robustness against jumps, since ``large'' jumps are relatively rare and are unlikely to fall in a narrow neighborhood of $\tau$ where $K_b$ places most of its mass.
For instance,
\cite[Theorem 13.3.3]{jacod2012discretization} shows that in the case of uniform kernels (i.e., $K(u)={\bf 1}_{[0,1)}(u)$ or $K(u)={\bf 1}_{(-1,0]}(u)$)
the estimator \eqref{KrnlSpotVol0} admits a rate-optimal CLT even in the case of unbounded variation jumps, provided $Y<4/3$\footnote{The case of jumps of bounded variation corresponds to $Y<1$, while the jumps are of unbounded variation when $Y>1$.}.  However, high levels of jump activity can still degrade performance, and the rate becomes suboptimal for  $Y\geq 4/3$:
in general, for any $a\in(0,1/2]$ such that $Y<2/(1+a)$, one can find a bandwidth $b_n$ such that $n^{a/2}(\hat\sigma_\tau^2-\sigma_\tau^2)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,V_\sigma)$, which yields a rate that is slower than $n^{-\frac{2-Y}{2Y}}$. Thus, if, for instance, $Y\leq 3/2$, the rate is slower than $n^{-1/6}$.
 Surprisingly, following the two-step approach \eqref{M1aTCLT}-\eqref{M1bTCLT}, even when thresholding is applied to \eqref{KrnlSpotVol0}, i.e.,
\begin{equation}\label{KrnlSpotVolThrholded}
    \hat\sigma^2_\tau=\sum_{i=1}^n K_b(t_{i-1}-\tau)(\Delta_i^nX)^2{\bf 1}_{\{|\Delta_i^nX|\leq v_n\}},
\end{equation}
there is no apparent asymptotic benefit:
the optimal rate of $n^{-1/4}$ remains attainable only if $Y<4/3$, and the best possible rate when $Y\geq 4/3$ remains bounded by $n^{-\frac{2-Y}{2Y}}$, which is strictly slower.

Given these limitations, one may wonder if estimation performance can be improved by modifying the kernel $K$. This question was addressed in \cite{Figueroa-Lopez_Wu_2024}, which extended the approach in \cite{jacod2012discretization} to two-sided kernels of unbounded support.  Although the convergence rate remains unchanged, they established that the asymptotic variance can be substantially reduced. For example, using the kernel $K(u)=.5e^{-|u|}$ leads to an asymptotic variance that is one-quarter of that obtained by a uniform kernel in the suboptimal rate regime $(Y\geq 4/3)$. Moreover, in the optimal rate regime $(Y< 4/3)$, this kernel was shown to be variance-optimal,  highlighting the utility of unbounded kernels in general. However, we note that the results in both \cite{jacod2012discretization} and \cite{Figueroa-Lopez_Wu_2024} are based on the two-step approach \eqref{M1aTCLT}-\eqref{M1bTCLT}, which we depart from in the present work.




 These findings motivate a fundamental question: \textit{can one develop methods that attain better rates -- possibly optimal --  beyond the boundary $Y< 4/3$}?
A recent step in this direction was taken by \cite{LIU2018}, who developed a localized variant of the characteristic function approach of \cite{Jacod2014}.
Assuming the infinite-variation jump component is of the form $\int_0^t\chi_{s}dL_s$, where $L$ is a strictly $Y$-stable symmetric L\'evy process, and while restricting to the class of compactly-supported  kernels  that are continuously differentiable, they establish that their estimator satisfies
\begin{equation}\label{LiuCLT0}
    \sqrt{\frac{b_n}{\Delta_n}}\frac{\hat\sigma^2_\tau-\sigma^2_\tau}{\sqrt{2\sigma_\tau}}\stackrel{\cal{D}}{\longrightarrow}\mathcal{N}\left(0,\int K^2(u)du\right),
\end{equation}
for any $Y\leq{}3/2$,  but with a suboptimal convergence rate (i.e., $\sqrt{\frac{\Delta_n}{b_n}}\gg n^{-1/4}$). By applying an additional debiasing step similar to that in \cite{Jacod2014}, they further showed that associated tuning parameters can be chosen so that \eqref{LiuCLT0} holds for any $Y<2$, however, still with a suboptimal rate and under the jump symmetry condition described above. Consequently, the boundary $Y<4/3$ has long marked the limit of efficiency in spot volatility estimation, and the regime $Y\geq 4/3$ has remained largely underexplored without symmetry assumptions.

In this work,  we push this efficiency boundary substantially forward by introducing appropriately debiased versions of the estimator \eqref{KrnlSpotVolThrholded}. First, by exploiting high-order expansions of the truncated moments of $\Delta_i X$, rather than relying on the coarser two-step framework  \eqref{M1aTCLT}-\eqref{M1bTCLT}, which does not explicitly account for higher-order bias,  we show the estimator \eqref{KrnlSpotVolThrholded} actually achieves rate-optimality for any $Y\leq{}3/2$ (namely, $n^{1/4}(\hat\sigma_\tau^2-\sigma_\tau^2)\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N}(0,V_\sigma)$) under standard assumptions and without additional adjustment. Second, by introducing debiasing steps similar to those in \cite{Jacod2014,BONIECE2024}, we construct fully efficient estimators that cover the range  $0<Y<20/11$. This extension is practically relevant, as empirical studies indicate that values of $Y$ are typically between $1.5$ and $1.8$ for liquid stocks (see \cite{BONIECE2024} for a brief supporting empirical analysis and further references). Our method stands in sharp contrast to the results in \cite{LIU2018}, offering rate-optimal estimators over a wider range of $Y$, broader applicability through more flexible model assumptions, reduced asymptotic variances enabled by the use of unbounded kernels, and simpler estimators that are easier to implement. To validate our results, we perform a comprehensive simulation study and find that our methods significantly outperform those of \cite{LIU2018} in terms of both bias and variance over finite samples. Our findings further demonstrate that the advantages of kernels with unbounded support extend to the debiasing framework developed here, providing substantial improvements over widely-used uniform kernels, which are often the default choice in practice, especially in inference problems that require estimation of spot volatility as a preliminary step.



The paper is organized as follows.  Section \ref{Sect2} introduces the setting, assumptions, and preliminary results. Section \ref{SectMainResults} presents the main results and their proofs.  Section \ref{SectionSimul} contains simulations assessing the performance of our estimators. Some additional technical proofs are collected in Appendix \ref{ApTechPrfs}.


\section{Setting and background}\label{Sect2}
Consider a 1-dimensional It\^o semimartingale $X = \big(X_t\big)_{t\in \mathbb{R}_{+}}$ defined on a complete filtered probability space $\big(\Omega,\mathscr{F},(\mathscr{F})_{t\in \mathbb{R}_{+}},\mathbb P\big)$ that can be decomposed as
\begin{align}\label{model}
    X_t =\int_0^tb_sds +  \int_0^t\sigma_s dW_s + \int_0^t\chi_{s^-}dJ_s +\int_0^t\int\delta(s,z)\mathfrak{p}(ds,dz),\;\;t\in [0,\infty),
\end{align}
where $W:=(W_t)_{t \in \mathbb{R}_{+}}$ is a Wiener process, $\sigma$ is an adapted càdlàg process, $J:=(J_t)_{t \in \mathbb{R}_{+}}$ is an independent pure-jump Lévy process with Lévy triplet $(b,0,v)$, and $\mathfrak{p}$ is a Poisson random measure on $\mathbb{R}_+\times\mathbb{R}$ with an intensity $\mathfrak{q}(ds,dz) = ds \otimes \lambda(dz)$, such that the $\lambda$ is a $\sigma$-finite measure on $\mathbb{R}$, possibly dependent with $J$. The Lévy measure $v$ is assumed to admit a density $s:{\mathbb{R}\backslash\{0\}}\rightarrow \mathbb{R}_+$ of the form
\begin{align}
    s(x):=\frac{dv}{dx}:=\big(C_+\mathbf{1}_{(0,\infty)}(x) + C_-\mathbf{1}_{(-\infty,0)}(x)\big)q(x)|x|^{-1-Y}.
\end{align}
{We have the following standing assumptions:}
\begin{asu}\label{BscAs1}\hfill
\begin{enumerate}
    \item  $C_{\pm}>0$ and  $Y \in (0,2)\setminus\{1\}$.
    \item $q:{{\mathbb{R}\backslash\{0\}} \rightarrow \mathbb{R}_+}$ is a bounded Borel-measurable function such that:
    \begin{enumerate}
        \item $q(x) \rightarrow 1$, as $x\rightarrow 0$.
        \item There exist $a_{\pm}$ such that
    \begin{equation}\label{CndqA}\int_0^1 \big|q(x)-1-a_+ x\big|x^{-Y-1}dx + \int_{-1}^0\big|q(x)-1-a_-x\big||x|^{-Y-1}dx<\infty.
        \end{equation}
    \end{enumerate}
    \item The process $\chi$ is given as
    \begin{equation}\label{CndChia}\chi_t = \chi_0 + \int_0^tb_x^\chi ds + \int_0^t \Sigma_s^\chi dB_s.
    \end{equation}
    \item The processes $W,B$ are standard Brownian motions with correlation $d\langle W,B\rangle_t = \rho_tdt$ for an adapted, locally bounded càdlàg process $\{\rho_t\}_{t\geq 0}$. $W,B$ are independent of $(J,\mathfrak{p})$;
    $J$ is independent of $\sigma$.
    \item The {processes} $\Sigma^\chi$, $b$, $b^\chi$ are càdlàg adapted, and {$\delta$} is predictable. There {exist} a sequence $\{\tau_n\}_{n\geq 1}$ of stopping times increasing to infinity, nonnegative {$\lambda(dz)$}-integrable function {$H$} and a positive sequence $\{M_n\}_{n\geq 1}$ such that
    \begin{equation*}
        t\leq \tau_n \Rightarrow \begin{cases}
            |\sigma_t| + |b_t| + |b_t^\chi| + |\Sigma_t^\chi| \leq M_n,\\
            ({|\delta(t,z)|}\wedge 1)^{{r}} \leq M_n {H(z)},
        \end{cases}
    \end{equation*}
    for some ${r}\in[0,1\wedge Y)$.
    \item The spot variance process $c_t:=\sigma_t^2$ is assumed to follow the dynamics:
\begin{align}\label{CndCa}
    c_t = c_0 + \int_0^t \tilde \mu_sds +\int_0^t\tilde \sigma_s dB_s,
\end{align}
where $B:=(B_t)_{t\geq 0}$ is as in point 3 above. Here $\{\tilde \mu\}_{t\geq 0}$ and $(\tilde\sigma_t)_{t\geq 0}$ are  adapted càdlàg locally bounded processes.
\end{enumerate}
\end{asu}



\begin{remark}\label{CommntCond}
The conditions above are essentially the same as those in \cite{BONIECE2024}. \cite{LIU2018} considered a similar but more restrictive component of unbounded variation by taking $J$ to be a symmetric strictly stable L\'evy process, while here we considered a type of tempered stable L\'evy process. The most technical (but still relatively mild) condition is that in \eqref{CndqA}, which is used to apply a density transformation or change of probability measure under which the process $J$ becomes stable. We refer to \cite{BONIECE2024} for further details.
\end{remark}

We assume we have at our disposal $n$ evenly-spaced discrete observations $X_{t_i}$ during a fixed time interval $[0,T]$. For simplicity, we further assume $T=1$ and, thus, $t_i=i/n$. Denote $\Delta_n = 1/n$, $\mathcal{F}_{i}^n:=\mathcal{F}_{t_i}$, and $\Delta^n_iA = A_{i\Delta_n} - A_{(i-1)\Delta_n}$ for any process $A$, where we often omit the superscript $n$. Our estimation target is {$c_\tau=\sigma_\tau^2$} for a fixed time $\tau\in(0,T)$.
We consider the spot volatility estimator
\begin{equation}\label{MEDHF}
\hat c_n(m_n, v_n) := \frac{\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)(\Delta_i^nX)^2\mathbf{1}_{\{|\Delta_i^nX|\leq  v_n\}}}{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}},
\end{equation}
where $K$ is a kernel function, and $m_n$ controls the  bandwidth {$b_n:=m_n\Delta_n$ such that} $m_n \rightarrow \infty$ and $m_n\Delta_n\rightarrow 0$, and  $K_b(x):=K(x/b)/b$. We assume the following conditions on $K$, which are standard:
\begin{asu}\label{asu_kernel}
   The kernel function $K:\mathbb{R}\rightarrow [0,\infty)$ is bounded and Lipschitz and piecewise $C^1$ on $(-\infty, \infty)$ such that $\int K(x)dx = 1$, $\int |K(x)x|dx<\infty$, $\int |K'(x)|dx<\infty$, and $K(y)y^2\rightarrow 0$ as $|y|\rightarrow \infty$.
\end{asu}
{The estimator \eqref{MEDHF} without the denominator factor, truncation, or any debiasing was studied in \cite{fan2008spot} and  \cite{kristensen2010nonparametric} (see also additional references in the introduction). The denominator, motivated by the standard Nadarayan-Watson nonparametric regression estimator, and noted in passing in \cite{kristensen2010nonparametric}, serves a dual role: it provides edge correction in finite samples and simplifies several proofs,  notably leading to more relaxed technical assumptions in the optimal bandwidth regime (see Remark \ref{SlyRmk1} below).  In contrast, when the denominator is omitted, our results remain valid, but stricter conditions on the bandwidth $b_n = m_n \Delta_n$ and its relationship with $Y$ are required; however, in the suboptimal rate regime (see Remark \ref{SlyRmk1}, case (ii), below), the denominator can be dispensed of entirely without any loss of generality.  The truncated version in \eqref{MEDHF} was studied in \cite{figueroa2020optimal} and \cite{Figueroa-Lopez_Wu_2024}, also without debiasing. Finally, Assumption \ref{asu_kernel} permits kernels of unbounded support, which, as discussed in the introduction, enables estimators that are more variance efficient in both the optimal and suboptimal convergence regimes.}

\medskip
\noindent
\textbf{Notation:} Throughout, {$A_n\lesssim B_n$ and $a_n\ll b_n$ mean $A_n = O_P(B_n)$ and $a_n=o(b_n)$, respectively.}  For any process $V$, when unambiguous we write $\Delta_iV$ in place of $\Delta_i^nV$.

\section{Main results}\label{SectMainResults}
The following moment expansions play key roles in our analysis. The proofs are based on arguments in analogous results in \cite{BONIECE2024} (see {Lemmas} 2, 3, 4 and 5 therein) and are given in Appendix \ref{FPNH0} for completeness.
\begin{proposition}\label{prop}
Suppose that $Y \in (0,2)\setminus\{1\}$. Let $\Delta_n^{\frac{1}{2}-s}\ll  v_n \ll \Delta_n^{\frac{1}{4-Y}}$ for a fixed $s\in(0,1/2)$. Then, the following statements hold:
\begin{enumerate}
    \item For any integer $p\geq 1$, we have
    \begin{align}\label{GNMExp}
        &\mathbb E\big((\Delta_iX)^{2 p}\mathbf{1}_{\{|\Delta_iX|\leq  v_n\}}|\mathcal F_{i-1}\big)=(2{p}-1)!!\sigma_{t_{i-1}}^{2{p}}\Delta_n^{ p}+C_{{p},i}\Delta_n v_n^{2{p}-Y}+o_P(\Delta_n v_n^{2{p}-Y})+o_P(\Delta_n^{p}),
    \end{align}
    where $C_{p,i} := \frac{(C_++C_-)|\chi_{t_{i-1}}|^Y}{2p-Y}$.
    \item Furthermore, if $p=1$, the following high-order expansion holds:
\begin{align}\label{PartExp0}
        &\mathbb E\big((\Delta_iX)^{2{}}\mathbf{1}_{\{|\Delta_iX|\leq  v_n\}}|\mathcal F_{i-1}\big)=\sigma_{t_{i-1}}^2\Delta_n+C_{1,i}\Delta_n v_n^{2-Y}+D_{1,i} \Delta_n^2 v_n^{-Y}+O_P(\Delta_n^3 v_n^{-2-Y}) +o_P(\Delta_n^{5/4}),
\end{align}
where $D_{1,i}=\frac{(C_++C_-)(Y+1)(Y+2)}{2Y}\sigma^2_{t_{i-1}}|\chi_{t_{i-1}}|^Y$. Additionally, {for any $\zeta>1$,} we have
\begin{equation}
\label{PartExp0_difference}
\begin{aligned}
\mathbb E\big((\Delta_iX)^{2{}}\mathbf{1}_{\{v_n<|\Delta_iX|\leq  \zeta v_n\}}|\mathcal F_{i-1}\big)
&=C_{1,i}\Delta_n(\zeta^{2-Y}-1) v_n^{2-Y}+D_{1,i} \Delta_n^2 (\zeta^{-Y}-1)v_n^{-Y}\\
&\quad
+O_P(\Delta_n^3 v_n^{-2-Y})
+o_P(\Delta_n^{\frac{3}{4}}v_n^{\frac{4-Y}{2}}).
\end{aligned}
\end{equation}
    \item  For any $p>1$ (not necessarily an integer), we have
\begin{align}\label{prop1_non_integer}
        &\mathbb E\big(|\Delta_iX|^{2p}\mathbf{1}_{\{|\Delta_iX|\leq  v_n\}}|\mathcal F_{i-1}\big) = O_P(\sigma_{t_{i-1}}^{2{p}}\Delta_n^{p})+O_P(C_{{p},t_{i-1}}\Delta_n v_n^{2{p}-Y}).
    \end{align}
\end{enumerate}
\end{proposition}
To analyze the asymptotic {behavior} of the estimator $\hat c_n(m_n, v_n)$ {defined in Eq.~\eqref{MEDHF}},  we consider the following decomposition:
\begin{align}\label{kernel_spot_estimator}
    \hat c_n(m_n, v_n)
    &=\left(\hat c_n(m_n, v_n) - \frac{\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2}{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}}\right) \nonumber\\
    &\quad+ \left(\frac{\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2}{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}}\right)\nonumber
    \\
    &=: \hat c_{n,1}(m_n, v_n) + \hat c_{n,2}(m_n).
\end{align}
Our next result is central to our analysis and underpins our main debiasing procedure. It establishes the joint asymptotic behavior of the terms in \eqref{kernel_spot_estimator}, accounting for appropriate bias terms, and provides the framework for separating qualitatively distinct asymptotic regimes for $m_n$.  In addition, on its own, it yields feasible CLTs for $\hat c_n(m_n,v_n)$ in certain settings that sharpen existing results in the literature (see Remark \ref{SlyRmk1}).
\begin{theorem}\label{clt}
    Suppose that $\Delta_n^{\beta}\ll  v_n \ll \Delta_n^{\beta'}$ with $\frac{1}{4-Y}<\beta'<\beta<\frac{1}{2}$, $\sqrt{\mathrm {log}(n)} \ll m_n\ll \Delta_n^{-4}v_n^{4+2Y}$, and $m_n = O(\Delta_n^{-\frac{1}{2}})$. Let
    \begin{align}
    \tilde Z_n(m_n, v_n) &:= \sqrt{m_n}\Big(\hat c_{n,1}(m_n, v_n) -A(v_n,m_n)\Big),\nonumber\\
    \tilde Z_n'(m_n) &:= \frac{1}{\sqrt{m_n\Delta_n}}\Big(\hat c_{n,2}(m_n) -\sigma^2_\tau\Big),\nonumber
\end{align}
where, using the notation in \eqref{PartExp0},
\begin{align}\label{AnTerm}
   {A(v_n,m_n) := \frac{\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau){C_{1,i}}\Delta_n v_n^{2-Y}}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} + \frac{\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau){D_{1,i}} \Delta_n^2 v_n^{-Y}}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}.}
\end{align}
Then, under Assumptions \ref{BscAs1} and \ref{asu_kernel}, as $n\rightarrow \infty$,
\begin{align}\label{clt_spot}
    (\tilde Z_n(m_n, v_n),\tilde Z'_n(m_n))
\xrightarrow{ st} \mathcal (Z_1,Z_2),
\end{align}
where $Z_1 \sim N\big(0,2\sigma_\tau^4\int K^2(x)dx\big)$ and $Z_2 \sim N(0, {\tilde\sigma^2_{\tau}}\int L^2(t) dt)$ with $L(t) = \int_t^{\infty}K(u)du\mathbf{1}_{t\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{t<0}$,
{and $Z_1$ and $Z_2$ are independent}.
\end{theorem}
\begin{remark}\label{SlyRmk1}
Taking $m_n\to\infty$ at different rates alters the balance of the errors appearing on the right-hand side of \eqref{clt_spot}.  Three distinct regimes can be distinguished:
(i) $m_n\sqrt{\Delta_n}\to\theta\in(0,\infty)$,
(ii) $m_n\sqrt{\Delta_n}\to 0$,
and (iii) $m_n\sqrt{\Delta_n}\to\infty$, which we remark on below.

\medskip
\noindent \textbf{Case (i): $m_n \sqrt{\Delta_n} \to \theta \in (0,\infty)$.} In this regime, the orders of $\sqrt{m_n}$ and $1/\sqrt{m_n\Delta_n}$ match, in which case Theorem \ref{clt} gives
    \begin{equation}\label{ITODelta}
        \Delta_n^{-1/4}\left(\hat c_n(m_n, v_n)-A(v_n,m_n)-\sigma_{\tau}^2\right)\xrightarrow{ st} \theta^{-1/2}Z_1+\theta^{1/2} Z_2,
    \end{equation}
or, equivalently,
    \begin{equation}\label{ITQRTm}
        \sqrt{m_n}\left(\hat c_n(m_n, v_n)-A(v_n,m_n)-\sigma_{\tau}^2\right)\xrightarrow{ st} Z_1+\theta Z_2.
    \end{equation}
Note that the bias term $A(v_n,m_n)$ is $O_P(v_n^{2-Y})$.  Thus, if  $\sqrt{m_n} v_n^{2-Y}\ll 1$, then \eqref{ITODelta} and \eqref{ITQRTm} remain true even with the term $A(v_n,m_n)$ omitted, in which case we obtain a valid feasible CLT for the estimation error $\hat c_n(m_n, v_n)-\sigma_{\tau}^2$ with convergence rate  $\Delta_n^{1/4}$ (which is optimal). While the condition $\sqrt{m_n} v_n^{2-Y}\ll 1$ can be met in principle by taking small enough $v_n$,  Theorem \ref{clt} additionally requires $v_n\gg \Delta_n^{1/2}$, and thus implicitly $\Delta_n^{-1/4}\Delta_n^{(2-Y)/2}\ll 1$, which can happen only if $Y<3/2$. Nevertheless, this result significantly improves upon \cite{Figueroa-Lopez_Wu_2024} (and also {\cite{jacod2012discretization}}), where a rate-optimal CLT is shown to be valid only when $Y<4/3$. Theorem \ref{clt} also yields stronger results than the characteristic function approach taken in \cite{LIU2018}, in which a similar CLT is is shown to be valid, under substantially more restrictive model assumptions\footnote{For instance, they assume that the process $J$ in \eqref{model} is a symmetric stable L\'evy process}, for $Y<3/2$ at a convergent rate strictly slower than $\Delta_n^{1/4}$ (due to an assumption that $m_n\sqrt{\Delta_n}\to{}0$).

Further, under case (i),  the conditions on $v_n$ take the form $\Delta_n^{\beta}\vee \Delta_n^{\frac{7}{4Y+8}}\ll v_n\ll \Delta_n^{\beta'}$ ($\frac{1}{4-Y}<\beta'<\beta<\frac{1}{2}$), which can hold only if $\frac{7}{4Y+8}>\frac{1}{4-Y}$, which requires $Y<\frac{20}{11}$. Thus, this rate-optimal case (i) covers nearly the entire admissible range of $Y$ observed in applications, and ultimately underpins our feasible efficient CLT in this range (Theorem \ref{main}).
\medskip

\noindent \textbf{Case (ii): $m_n \sqrt{\Delta_n} \to  0$.}  This regime corresponds to setting $\theta=0$ in \eqref{ITQRTm}. Indeed, \begin{equation}\label{ITQRTmC2}
        \sqrt{m_n}\left(\hat c_n(m_n, v_n)-A(v_n,m_n)-\sigma_{\tau}^2\right)
        =\tilde Z_n(m_n, v_n)+m_n\sqrt{\Delta_n}\tilde Z'_n(m_n, v_n),
    \end{equation}
    and, from \eqref{clt_spot}, the first term above converges to $Z_1$, while the second is $o_P(1)$. In this case, the attained convergence rate, $1/\sqrt{m_n}$, is suboptimal, i.e., slower than the $\Delta^{1/4}$ rate achieved when $\theta\in(0,\infty)$.  However,  this asymptotic regime has the practical advantage that any eventual estimation of the volatility-of-volatiltity $\tilde\sigma_\tau^2$ is not needed when constructing confidence intervals, since $Z_2$ is absent from the limit. If we also require $\sqrt{m_n} v_n^{2-Y}\ll 1$, then the bias $A(v_n,m_n)$ can be omitted in \eqref{ITQRTmC2} leading to a feasible CLT for $\hat c_n(m_n, v_n)$ centered at $\sigma_{\tau}^2$.  Under this constraint, the condition  $v_n\gg \Delta_n^{\frac{1}{2}}$ and the asymptotic condition of case (ii) together imply that the convergence rate $m_n^{-1/2}$ must be strictly slower than $\Delta_n^{\frac14 \wedge \frac {2-Y}2}$, yielding a convergence rate that can be taken arbitrarily close to $\Delta_n^{1/4}$  when $Y<3/2$ but deteriorates as $Y$ increases to $2$.
    \medskip

\noindent \textbf{Case (iii): $m_n \sqrt{\Delta_n} \to \infty$.}
    Though technically excluded from our hypotheses, our results can also cover the regime $m_n\sqrt{\Delta_n}\to\infty$, but at the expense of an upper bound of $Y< 8/5$ in the jump activity index. In this setting, the attained convergence rate is $\sqrt{m_n\Delta_n}$, which is not as fast as the analogous rates when $\theta\in[0,\infty)$.
\end{remark}



\begin{remark} Beyond finite-sample edge correction, the denominator in \eqref{MEDHF} (which converges to 1) relaxes the technical conditions required for our results; without it, additional restrictions on
$Y$ and $m_n$ are necessary.
   Specifically, define ${\tilde c_n(m_n,
   v_n)} := \sum_{i=1}^{n} K_{m_n\Delta_n}(t_{i-1}-\tau)(\Delta_iX)^2 \mathbf{1}_{\{|\Delta_iX|\leq v_n\}}$, and let $\tilde Z_n(m_n, v_n)$, $\tilde Z'_n(m_n)$, and $A(v_n,m_n)$ denote the corresponding analogs without the denominator. Then the limit \eqref{clt_spot} still holds under the hypotheses of Theorem~\ref{clt}, provided also that $m_n \gg \Delta_n^{-1/3}$. Under this additional condition on $m_n$, \eqref{ITQRTm} remains valid for $Y<20/11$ when $m_n \sqrt{\Delta_n} \to \theta \in (0,\infty)$. However, a much more substantive difference arises when $m_n \sqrt{\Delta_n} \to 0$: if we also impose $\sqrt{m_n} v_n^{2-Y}\ll 1$ to obtain a feasible CLT (as discussed in Remark~\ref{SlyRmk1}), the constraint becomes $\Delta_n^{-1/3} \ll m_n \ll v_n^{2(Y-2)} \ll \Delta_n^{Y-2}$, which can only be satisfied if $Y<5/3$, illustrating the benefit of including the denominator factor in \eqref{MEDHF}.
\end{remark}



\begin{proof}[Proof of Theorem \ref{clt}]
Denote
\begin{align*}
Y_i &:= \sqrt{m_n}K_{m_n\Delta_n}(t_{i-1}-\tau)(\Delta_iX)^2\mathbf{1}_{\{|\Delta_iX|\leq  v_n\}},\\
    Y'_i &:= \frac{\Delta_n}{\sqrt{m_n\Delta_n}}\times\left\{
    \begin{aligned}
        &0, \;&\text{if } i=1,\\
        -&\Big(\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\Delta_iB,\; &\text{if } 2\leq i\leq \ceil{\tau/\Delta_n},\\
       &\Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\Delta_iB,\; &\text{if } \ceil{\tau/\Delta_n} < i \leq n.
    \end{aligned}\right.
\end{align*}
Consider the following decompositions:
\begin{align}\label{TrmT1T2}
    \tilde Z_n(m_n, v_n)
    &=\Big(\frac{\sum_{i=1}^n Y_i}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} - \frac{\sum_{i=1}^n \mathbb E(Y_i|\mathcal{F}_{i-1})}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}\Big) \nonumber\\
    &\qquad+ \Big(\frac{\sum_{i=1}^n \mathbb E(Y_i|\mathcal{F}_{i-1})}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} - \sqrt{m_n}\Delta_n\sum_{i=1}^n\frac{K_{m_n\Delta_n}(t_{i-1}-\tau)}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}\sigma_{t_{i-1}}^2 -\sqrt{m_n}A(v_n,m_n)\Big)\nonumber\\
    &=:T_1 + T_2,
\end{align}
and
\begin{align}\label{TrmT3T4a}
    \tilde Z'_n(m_n) &= \tilde \sigma_{\tau}\frac{\sum_{i=1}^nY_i'}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} + \Big({\tilde Z'_n(m_n)}- \frac{\tilde\sigma_{\tau}\sum_{i=1}^n Y_i'}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}
    \Big)\nonumber\\
    &=\tilde\sigma_{\tau} \frac{\sum_{i=1}^nY_i'}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}  \nonumber\\
    &\quad+ \Big(\frac{\Delta_n}{\sqrt{m_n\Delta_n}}\sum_{i=1}^{n}\frac{K_{m_n\Delta_n}(t_{i-1}-\tau)(\sigma^2_{t_{i-1}} - \sigma^2_\tau)}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} - \tilde\sigma_{\tau} \frac{\sum_{i=1}^nY_i'}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}\Big)\nonumber\\
    &=:T_3 + T_4.
\end{align}
We now proceed to show $(T_1,T_3)\xrightarrow{st} (Z_1, \tilde{\sigma}_\tau Z_2)$ by verifying the conditions of Theorem 2.2.15 in \cite{jacod2012discretization}, and in addition that $T_2$ and $T_4$ are asymptotically negligible.

\medskip
\noindent
{\bf a)} We start with $T_3$. Clearly, $\mathbb E(Y'_i|\mathcal F_{i-1})  = 0$ and, using the moment formula for Gaussian distributions {and Lemma 3.1 in \cite{FIGUEROALOPEZ20204693}}, we have
\begin{align*}
    \sum_{i=1}^n {\mathrm{Var}(Y'_i|\mathcal F_{i-1})}&= \frac{\Delta_n^2}{m_n\Delta_n} \sum_{i=1}^n \left(\Big(-\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)^2\Delta_n\mathbf{1}_{\tau\geq t_i} + \Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)^2\Delta_n\mathbf{1}_{\tau< t_i}\right)\nonumber\\
    &\quad=  \frac{1}{m_n\Delta_n}\int_0^1 \Big(\int_{{v}}^{1}K_{m_n\Delta_n}({s}-\tau)ds\mathbf{1}_{\tau\geq {v}}-\int^{{v}}_{0}K_{m_n\Delta_n}({s}-\tau)ds\mathbf{1}_{\tau< {v}}\Big)^2 dv + o_P(1)\nonumber\\
    &\quad \xrightarrow{P} \int \Big(\int_t^{\infty}K(u)du\mathbf{1}_{t\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{t<0}\Big)^2 dt,\nonumber
\end{align*}
and
 \begin{align*}
    \sum_{i=1}^n\mathbb E({Y'}_i^4|\mathcal F_{i-1})&= \frac{3\Delta_n^4}{m_n^2\Delta_n^2} \sum_{i=1}^n \left(\Big(\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)^4\Delta_n^2\mathbf{1}_{\tau\geq t_i} + \Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)^4\Delta_n^2\mathbf{1}_{\tau< i}\right)\nonumber\\
    &\quad =\frac{3}{m_n}\int \left(\int_t^{\infty}K(u)du\mathbf{1}_{t\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{t<0}\right)^4 dt + o_P(1)\xrightarrow{P} 0.\nonumber
\end{align*}
On the other hand, by Assumption \ref{asu_kernel} and Lemma 3.1 in \cite{FIGUEROALOPEZ20204693}, with $f\equiv 1$ and $m=1$ therein, we have
\begin{align}\label{kernel_convergence}
    \Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau) - 1 =
    {O(m_n^{-1})},
\end{align}
which vanishes as $n\rightarrow \infty$. Therefore,
\begin{gather*}
\frac{\sum_{i=1}^n {\mathrm{Var}(Y'_i|\mathcal F_{i-1})}}{{\big(\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)\big)^2}} \xrightarrow{P} \int \Big(\int_t^{\infty}K(u)du\mathbf{1}_{t\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{t<0}\Big)^2 dt,\\
\text{and }\;\frac{\sum_{i=1}^n\mathbb E({Y'}_i^4|\mathcal F_{i-1})}{\big(\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)\big)^4}\rightarrow 0.
\end{gather*}
To conclude $T_3 \xrightarrow{ st}  \tilde{\sigma}_\tau Z_2$,
from  Theorem 2.2.15 in \cite{jacod2012discretization} it suffices to show the following technical condition:
\begin{align}\label{initial_technicalYprime}
    \sum_{i=1}^n\mathbb E_{i-1}\big(Y'_i\Delta_iM\big) \rightarrow 0,
\end{align}
where $\mathbb E_{i-1}(\cdot) = \mathbb E(\cdot|\mathcal{F}_{i-1})$ and $M$ is either $W$ or $B$ or is in the set $\mathcal N$ containing all bounded martingales orthogonal to $W$ and $B$.
When $M = B$,
we have
\begin{align*}
    \sum_{i=1}^n\mathbb E(Y'_i\Delta_iB|\mathcal F_{i-1})
    &= \frac{\Delta_n}{\sqrt{m_n\Delta_n}} \sum_{i=1}^n \left(-\Big(\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\Delta_n\mathbf{1}_{\tau\geq t_i} + \Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\Delta_n\mathbf{1}_{\tau<{ t_i}}\right)\nonumber\\
    & =  \frac{1}{\sqrt{m_n\Delta_n}}\int_0^1 \Big(\int_{{v}}^{1}K_{m_n\Delta_n}({s}-{\tau})ds\mathbf{1}_{\tau\geq {v}}-\int^{{v}}_{0}K_{m_n\Delta_n}(s-{\tau})ds\mathbf{1}_{\tau<{v}}\Big) d{v} +  o(1)\nonumber\\
    & =\sqrt{m_n\Delta_n} \int \Big(\int_t^{\infty}K(u)du\mathbf{1}_{t\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{t<0}\Big) d{t} +  o(1) \xrightarrow{P} 0.
\end{align*}
When $M \in \mathcal N$ or $M = W$, since $
\mathbb E(\Delta_iB\Delta_iW|\mathcal{F}_{i-1}) = O_P(\Delta_n)$, similar to the case with $M=B$, we obtain
$$\sum_{i=1}^n\mathbb E\big(Y'_i\Delta_iM|\mathcal{F}_{i-1}\big)\lesssim \frac{1}{\sqrt {m_n\Delta_n}}\int \Big(\int_t^{\infty}K(u)du\mathbf{1}_{\tau\geq 0}-\int^t_{-\infty}K(u)du\mathbf{1}_{\tau<0}\Big)dt \rightarrow 0.$$

\medskip
\noindent
{\bf b)}
Next, we work with the term $T_4$ of \eqref{TrmT3T4a}.
Let $i'\in\{1,\dots,n\}$ be such that $\tau \in (t_{i'-1},t_{i'}]$. Notice that
\begin{align}
    \sum_{i=1}^n Y_i' &= -\sum_{l=1}^{i'-1}\big(\sum_{i=l+1}^{i'}\Delta_iB\big)K_{m_n\Delta_n}(t_{l-1}-\tau) + \sum_{l=i'+1}^n\big(\sum_{i=i'+1}^{l}\Delta_iB\big)K_{m_n\Delta_n}(t_{l-1}-\tau)\\
    &=\sum_{l=1}^{i'-1}K_{m_n\Delta_n}(t_{l-1}-\tau)(B_{t_l}-B_{t_{i'}}) + \sum_{l=i'+1}^nK_{m_n\Delta_n}(t_{l-1}-\tau)(B_{t_l}-B_{t_{i'}}).
\end{align}
{In light of \eqref{kernel_convergence}}, we can then write
\begin{align}
    |T_4|
    &\lesssim \Big|\frac{1}{\sqrt{m_n\Delta_n}}\Big(\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\big(\int_{\tau}^{t_{i-1}}(\tilde\sigma_s-\tilde\sigma_\tau)dB_s+\int_{\tau}^{t_{i-1}}\tilde \mu_sds\big)\Big)\Big|\nonumber\\
    &\quad + \Big| \frac{1}{\sqrt{m_n\Delta_n}} \Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\Big(\int_{t_{i-1}}^{t_{i}}\tilde\sigma_\tau dB_s+\int_{\tau}^{t_{i'}}\tilde\sigma_\tau dB_s\Big)\Big|=:|T_{4,1}|+|T_{4,2}|.
\end{align}
Since $|B_{t_{i'}} - B_\tau| = O_P(\Delta_n^{1/2})$ and $\sup_i|B_{t_i}-B_{t_{i-1}}| = O_P(\Delta_n^{1/2}\mathrm {log}(n)^{1/2})$, the second term $T_{4,2}$ has order $O_P(\frac{\sqrt{\Delta_n\mathrm {log}(n)}}{\sqrt{m_n\Delta_n}})$ and vanishes given that $m_n\gg \sqrt{\mathrm {log}(n)}$ as $n\rightarrow \infty$. We just need to worry about the first term.
Let
\[
    \eta_i := \int_{\tau}^{t_{i-1}}(\tilde\sigma_\tau-\tilde\sigma_s)dB_s+\int_{\tau}^{t_{i-1}}\tilde \mu_sds,\quad\text{and}\quad
    \quad \rho(i) := \frac{1}{t_{i-1}-\tau}\mathbb E\Big(\int_{\tau}^{t_{i-1}}|\tilde\sigma_s - \tilde \sigma_{\tau}|^2ds\Big).
\]
Note that, following the same argument as in the proof of (13.3.37) for $j=6$ in \cite{jacod2012discretization} (see also the proof of (A.6) for $l=4$ in \cite{Figueroa-Lopez_Wu_2024}),
and using that $\tilde \sigma$ is càdlàg and bounded, for any positive sequence $N_n \rightarrow \infty$ with  $m_n\Delta_nN_n\rightarrow 0$ as $n\rightarrow \infty$, it holds that
$$\mathbb E(\eta_i^21_{i\in\mathcal{I}_n})\leq c|t_{i-1}-\tau|\rho(i)\quad \text{and}\quad\sup_{i\in\mathcal{I}_n}\rho(i)  \rightarrow 0,$$
where $\mathcal{I}_n:=\{i\in\{1,\dots,n\}:t_i \in (\tau-Nm_n\Delta_n,\tau + Nm_n\Delta_n)\}$.
Additionally, using the It\^o isometry and boundedness of $\tilde{\sigma}$ and $\tilde \mu$,  it is easily seen that $\mathbb E \eta_i^2 \lesssim |\tau - t_i|$ for any $i$. Thus,
\begin{align*}
    \mathbb E|T_{4,1}|
     &\lesssim\frac{1}{\sqrt{m_n\Delta_n}}\Big(\Delta_n\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau){\left(\mathbf{1}_{i\in\mathcal{I}_n}+
     \mathbf{1}_{i\notin\mathcal{I}_n}\right)}\sqrt{\mathbb E\eta_i^2}\Big)\\
     &\lesssim\frac{1}{\sqrt{m_n\Delta_n}}\Big(\Delta_n\sum_{i=1}^{n}\frac{1}{m_n\Delta_n}\mathbf{1}_{i\in\mathcal{I}_n}\sqrt{|t_{i-1}-\tau|\rho_i}\Big)\\
    &\quad+ \frac{1}{\sqrt{m_n\Delta_n}}\Big(\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau))\mathbf{1}_{i\notin\mathcal{I}_n}\sqrt{|t_{i-1}-\tau|}\Big) \\
    & \lesssim \frac{m_n\Delta_n}{\sqrt{m_n\Delta_n}m_n\Delta_n}N_n\sqrt{N_n m_n\Delta_n \sup_{i\in\mathcal{I}_n}\rho_i} +\int_{|u|>N} K(u)\sqrt{|u|} du\Big).
\end{align*}
Then, we see that $T_4=o_P(1)$ by taking $N_n\to\infty$ slow enough such that $N_n^{3/2}\sup_{i\in\mathcal{I}_n}\rho_i \rightarrow 0$
(such an $N_n$ is possible since as $N_n$ diverges more slowly, the quantity $\sup_{i\in\mathcal{I}_n}\rho_i$ vanishes at a faster rate).
Now, we conclude
\begin{align*}
    \frac{1}{\sqrt{m_n\Delta_n}}\tilde Z'_n(m_n) \rightarrow  N\big(0,\tilde\sigma_{\tau}^2 L^2(t)\big).
\end{align*}



\medskip
\noindent
{\bf c)}
We now proceed to consider $T_1$ and $T_2$ of Eq.~\eqref{TrmT1T2}. First, by Proposition \ref{prop}, for any $p>2$, we can write
\begin{align*}
    \mathbb E(Y_i|\mathcal F_{i-1}) & = \sqrt{m_n}K_{m_n\Delta_n}(t_{i-1}-\tau)\big(\sigma_{t_{i-1}}^2\Delta_n+C_{1,i}\Delta_n v_n^{2-Y}+D_{1,i} \Delta_n^2 v_n^{-Y}+O_P(\Delta_n^3 v_n^{-2-Y}) +o_P(\Delta_n^{5/4})\big),\nonumber\\
    \mathbb E(Y_i^p|\mathcal F_{i-1}) & \lesssim m_n^{p/2} K^p_{m_n\Delta_n}(t_{i-1}-\tau)\big(\Delta_n^{p}\sigma_{t_{i-1}}^{2p} + \Delta_n v_n^{2p-Y}\big),\nonumber\\
    \mathrm{Var}(Y_i|\mathcal F_{i-1}) &=m_n K^2_{m_n\Delta_n}(t_{i-1}-\tau)\big( 2\Delta_n^2\sigma_{t_{i-1}}^4 + O_P(\Delta_n^2 v_n^{2-Y}) + O(\Delta_n v_n^{4-Y})\big) \nonumber\\
    &=m_n K^2_{m_n\Delta_n}(t_{i-1}-\tau)\big( 2\Delta_n^2\sigma_{t_{i-1}}^4 + O_P(\Delta_n v_n^{4-Y})\big).
\end{align*}
Then,
$$s_{1n}^2 :=\sum_{i=1}^n\mathrm{Var}(Y_i|\mathcal F_{i-1}) = m_n\sum_{i=1}^nK^2_{m_n\Delta_n}(t_{i-1}-\tau)\big( 2\Delta_n^2\sigma_{t_{i-1}}^4 + O_P(\Delta_n v_n^{4-Y})\big).$$
Given $ v_n \ll \Delta_n^{\frac{1}{4-Y}}$ and \eqref{kernel_convergence}, we obtain
\begin{align*}
    \frac{s_{1n}^2}{(\Delta_n\sum_{j=1}K_{m_n\Delta_n}(t_{j-1}-\tau))^2}\rightarrow2\sigma^4_\tau\int K^2(x)dx.
\end{align*}
Moreover, we have
\begin{align*}
    \sum_{i=1}^n\mathbb E(Y_i^p|\mathcal F_{i-1}) \lesssim \frac{1}{m_n^{p/2-1}\Delta_n^{p-1}}\int K^p(x)dx (\sigma^{2p}_{\tau}\Delta_n^{p-1} + O_P(v_n^{2p-Y})).
\end{align*}
If we rewrite $p = (2 \ell +1)/ \ell$ for some ${\ell}>0$, the right-hand side of the above inequality vanishes if and only if
\begin{align*}
    m_n^{-\frac{1}{2\ell}}\Delta_n^{-1-\frac{1}{\ell}}v_n^{\frac{4\ell+2}{\ell}-Y} = \Big((\Delta_n^{-1}v_n^{4-Y})^\ell (m_n^{-\frac{1}{2}}\Delta_n^{-1}v_n^{2})\Big)^{\frac{1}{\ell}} = \Delta_n^{-1}v_n^{4-Y}  \Big(m_n^{-\frac{1}{2}}\Delta_n^{-1}v_n^{2}\Big)^{\frac{1}{\ell}}\rightarrow 0.
\end{align*}
Since $\Delta_n^{-1}v_n^{4-Y}\ll \Delta_n^{\beta'(4-Y)-1}\ll \Delta_n^s$ for small enough $s>0$, the above vanishes by picking $\ell$ large enough.
In light of \eqref{kernel_convergence}, we also have $\frac{\sum_{i=1}^n\mathbb E(Y_i^p|\mathcal F_{i-1})}{(\Delta_n\sum_{j=1}K_{m_n\Delta_n}(t_{j-1}-\tau))^p}\rightarrow 0$ as $n\rightarrow \infty$ for such {an $\ell$}.
To show  $T_2=o_P(1)$, recall $m_n = O(\Delta_n^{-\frac{1}{2}})$ and, thus, by Proposition \ref{prop} and \eqref{kernel_convergence},
\begin{align}
   \sum_{i=1}^n\mathbb E(Y_i|\mathcal F_{i-1}) &=\sqrt{m_n}\sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau)\Delta_n\sigma_{t_{i-1}}^2 +\sqrt{m_n}\sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau)C_{1,i} \Delta_nv_n^{2-Y} \nonumber\\
    &+ \sqrt{m_n}\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau)D_{1,i} \Delta_n^2 v_n^{-Y} + O_P(\sqrt{m_n}\Delta_n^2 v_n^{-2-Y})+o_P(1).
\end{align}
Hence,
\begin{align*}
T_2&=\frac{\sum_{i=1}^n \mathbb E(Y_i|\mathcal{F}_{i-1})}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} - \sqrt{m_n}\Delta_n\sum_{i=1}^n\frac{K_{m_n\Delta_n}(t_{i-1}-\tau)}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}\sigma_{t_{i-1}}^2 -\sqrt{m_n}A(v_n,m_n)\\
&= O_P(\sqrt{m_n}\Delta_n^2 v_n^{-2-Y}),
\end{align*}
which vanishes due to the condition $m_n\ll \Delta_n^{-4}v_n^{4+2Y}$. To obtain $T_1\xrightarrow{st} Z_1$, by Theorem 2.2.15 in \cite{jacod2012discretization}, it suffices to check the condition:
\begin{align}\label{initial_technical}
    \sum_{i=1}^n\mathbb E_{i-1}\big(Y_i\Delta_iM\big) \rightarrow 0,
\end{align}
where $M$ is either $W$ or $B$ or is in the set $\mathcal N$ containing all bounded martingales orthogonal to $W$ and $B$. The proof of \eqref{initial_technical} is technical and is deferred to Appendix \ref{TecnPrf1}.
This concludes the proof for $\tilde Z_n(m_n, v_n)
\xrightarrow{ st} Z_1$.


\medskip
\noindent
\textbf{d)}
To establish the joint convergence \eqref{clt_spot} and the asymptotic independence of $T_1,T_2$, it is sufficient to show that
\begin{align}\label{cov_summand}
    \sup_i\Big|\mathbb E\big((\Delta_iX)^2\mathbf{1}_{\{|\Delta_iX|\leq v_n\}}\Delta_iB|\mathcal{F}_{i-1}\big)\Big| = o_P(\Delta_n^{3/2}),
\end{align}
which is shown in Appendix \ref{TecnPrf1}. Indeed, from \eqref{cov_summand} we obtain:
    \begin{align*}
        &\sum_{i=1}^n\Big|\mathbb E(\left. Y_iY'_i\right|\mathcal F_{i-1})\Big|\nonumber\\
        &\quad = \frac{\sqrt{m_n}\Delta_n}{\sqrt{m_n\Delta_n}}\sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau)\cdot\left|\Big(-\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\mathbf{1}_{\tau\geq t_i} + \Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\mathbf{1}_{\tau< t_i}\right|\nonumber\\
        &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\Big|\mathbb E\big((\Delta_iX)^2\mathbf{1}_{\{|\Delta_iX|\leq v_n\}}\Delta_iB|\mathcal{F}_{i-1}\big)\Big|\nonumber\\
        &\quad = \Delta_n^2\sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau)\cdot\left(\Big(\sum_{l=1}^{i-1}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\mathbf{1}_{\tau\geq t_i} + \Big(\sum_{l=i}^{n}K_{m_n\Delta_n}(t_{l-1}-\tau)\Big)\mathbf{1}_{\tau< t_i}\right)o_P(1)\nonumber\\
        &\quad = \int_0^1 K_{m_n\Delta_n}(v-\tau)\Big({\int_{0}^{v}}K_{m_n\Delta_n}(s-\tau)ds\mathbf{1}_{\tau\geq  v}+{\int_{ v}^{1}}K_{m_n\Delta_n}( s-\tau)ds\mathbf{1}_{\tau< v}\Big) dv \cdot o_P(1)\xrightarrow{P} 0.
    \end{align*}
    In view of \eqref{initial_technicalYprime} and \eqref{initial_technical}, one can finally apply Theorem 2.2.15 in  \cite{jacod2012discretization} to conclude the proof.
\end{proof}

The next result is the second key ingredient needed for our debiasing method.  For an arbitrary fixed $\zeta>1$, it establishes the convergence rate of the difference $\tilde Z_n(m_n, \zeta v_n)-\tilde Z_n(m_n, v_n)$, i.e. of $\hat c_{n,1}(m_n,\zeta v_n)-\hat c_{n,1}(m_n, v_n)$ subject to a  bias correction, which we exploit in our debiasing procedure.
\begin{theorem}\label{thm_clt_difference}
Suppose that $1<Y<2$, $ \sqrt{\mathrm {log}(n)} \ll m_n$, $m_n = O(\Delta_n^{-\frac{1}{2}})$,  $\Delta_n^{\beta}\ll  v_n \ll \Delta_n^{\beta'}$ with $\frac{1}{4-Y}<\beta'<\beta<\frac{1}{2}$, and $m_n \rightarrow \infty$ such that $v_n^{\gamma}\Delta_n^{-1}\ll m_n \ll \Delta_n^{-5} v_n^{8+Y}$ with some $\gamma<Y$.
Then, under Assumptions \ref{BscAs1} and \ref{asu_kernel}, for an arbitrary fixed $\zeta > 1$, we have
\begin{align}\label{clt_difference}
    u_n^{-1}\big(\tilde Z_n(m_n, \zeta v_n)-\tilde Z_n(m_n, v_n)\big)
\xrightarrow{\mathcal D} \mathcal N
\left(0,\frac{(C_++C_-)|\chi_{\tau}|^Y}{4-Y}(\zeta^{4-Y}-1)\int K^2(x)dx\right),
\end{align}
as $n\rightarrow \infty$, where $u_n:= \Delta_n^{-1/2} v_n^{2-Y/2}\rightarrow 0$.
\end{theorem}

\begin{remark}\label{INTNTL}
The conditions on $m_n$ in Theorem \ref{thm_clt_difference} are stricter than those in Theorem \ref{clt} due to the condition $\Delta_n^{-1}v_n^\gamma\ll m_n\ll \Delta_n^{-5}v_n^{8+Y}$ (note $\Delta_n^{-5}v_n^{8+Y} \ll \Delta_n^{-4}v_n^{4+2Y}$, which was the the upper bound for $m_n$ in Theorem \ref{clt}).
As we shall see, the optimal rate of convergence of the estimation error of our ultimate debiased estimator can be achieved by taking $m_n=\theta\Delta_n^{-1/2}$ for any constant $\theta>0$ (cf.~Remark \ref{SlyRmk1}). In this bandwidth regime, the additional constraint  on $m_n$ in Theorem \ref{thm_clt_difference} reduces to $\Delta_n^{\frac{9}{16+2Y}}\ll  v_n \ll  \min(\Delta_n^{\frac{1}{4-Y}},\Delta_n^{\frac{1}{2\gamma}})$, whose upper bound and lower bound {can be made} compatible only if ${Y}<20/11$.
\end{remark}






\begin{proof}
    Denote
$$Z_i =  \frac{m_n^{1/2}}{\Delta_n^{-1/2} v_n^{2-Y/2}} K_{m_n\Delta_n}(t_{i-1}-\tau)(\Delta_iX)^2\left(\mathbf{1}_{\{|\Delta_iX|\leq \zeta  v_n\}}-\mathbf{1}_{\{|\Delta_iX|\leq  v_n\}}\right),$$
and note that
\begin{align*}
u_n^{-1}\big(\tilde Z_n(m_n, \zeta v_n)-\tilde Z_n(m_n, v_n)\big)&=\frac{\sum_{i=1}^n Z_i}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}-{u_n^{-1}\sqrt{m_n}\left(A(\zeta v_n,m_n)-A(v_n,m_n)\right)}\\
&=:\frac{\sum_{i=1}^n\xi_i}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}\\
&\quad+\Big(\frac{\sum_{i=1}^n\mathbb E(Z_i|\mathcal{F}_{i-1})}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}-u_n^{-1}\sqrt{m_n}\left(A(\zeta v_n,m_n)-A(v_n,m_n)\right)\Big),
\end{align*}
{where $\xi_i:=Z_i-\mathbb E(Z_i|\mathcal{F}_{i-1})$}. The second term above is $o_P(1)$. Indeed, since $m_n = O(\Delta_n^{-\frac{1}{2}})$, {by \eqref{PartExp0_difference} in Proposition \ref{prop},
\begin{align*}
    &\frac{\sum_{i=1}^n\mathbb E(Z_i|\mathcal{F}_{i-1})}{\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)} - u_n^{-1}\sqrt{m_n}\big(A({\zeta v_n},m_n) - A({v_n},m_n)\big)\\
    &\quad\lesssim \frac{m_n^{1/2}\Delta_n^2 v_n^{-2-Y}}{\Delta_n^{-1/2} v_n^{2-Y/2}}
    +\frac{m_n^{1/2}o_P(\Delta_n^{-1/4} v_n^{2-Y/2})}{\Delta_n^{-1/2} v_n^{2-Y/2}}\\
    &\quad= m_n^{1/2}\Delta_n^{5/2}v_n^{-4-Y/2}+o_P(1),
\end{align*}
which} vanishes given our assumption $m_n \ll \Delta_n^{-5}v_n^{8+Y}$.  Next we obtain the asymptotic behavior for $\sum_{i=1}^n {\xi_i}$ by verifying the conditions of Theorem 2.2.15 in \cite{jacod2012discretization}.
Expression \eqref{GNMExp} in Proposition \ref{prop} yields
\begin{align*}
    \mathrm{Var}({\xi_i}|\mathcal{F}_{i-1}) & = \frac{m_n}{\Delta_n^{-1} v_n^{4-Y}} K^2_{m_n\Delta_n}(t_{i-1}-\tau)\Big((\zeta^{4-Y}-1){C_{2,i}}\Delta_n v_n^{4-Y} + {o_P(\Delta_n v_n^{4-Y})}\\
     &\qquad\qquad\quad\qquad\qquad\qquad\qquad- \big\{(\zeta^{2-Y}-1){C_{1,i}}\Delta_nv_n^{2-Y}+(\zeta^{-Y}-1){D_{1,i}}\Delta_n^2v_n^{-Y}+{O_P(\Delta_n^3 v_n^{-2-Y})}\big\}^2 \Big)\nonumber\\
    & = \frac{m_n}{\Delta_n^{-1} v_n^{4-Y}} K^2_{m_n\Delta_n}(t_{i-1}-\tau)\big( (\zeta^{4-Y}-1){C_{2,i}}\Delta_n v_n^{4-Y} + o_P(\Delta_n v_n^{4-Y})
    \big),
\end{align*}
where $C_{2,i} = \frac{(C_++C_-)|\chi_{t_{i-1}}|^Y}{4-Y}$.
{Therefore,} with $1<Y<2$ and $ v_n \ll \Delta_n^{1/(4-Y)}$, we obtain
\begin{align*}
    s_{2n}^2 &:= \sum_{i=1}^n \mathrm{Var}({\xi_i}|\mathcal{F}_{i-1}) \\
    &=m_n\Delta_n^2 \sum_{i=1}^nK^2_{m_n\Delta_n}(t_{i-1}-\tau)\frac{(C_++C_-)|\chi_{t_{i-1}}|^Y}{4-Y}(\zeta^{{4-Y}}-1) + o_P(1) \\
    &\longrightarrow
    \frac{(C_++C_-)|\chi_{\tau}|^Y}{4-Y}(\zeta^{{4-Y}}-1)\int K^2(x)dx.
\end{align*}
And consequently, with \eqref{kernel_convergence}, we have
\begin{equation}\label{th2limitvar}
\frac{s_{2n}^2}{\big(\Delta_n\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)\big)^2} \rightarrow \frac{(C_++C_-)|\chi_{\tau}|^Y}{4-Y}(\zeta^{{4-Y}}-1)\int K^2(x)dx.
\end{equation}
{Similarly, for any} $p>2$, we have\footnote{{By a standard localization argument, we may assume that $\sigma$ is bounded.}}
\begin{align}
    \frac{\sum_{i=1}^n\mathbb E({\xi_i^p}|\mathcal{F}_{i-1})}{\big(\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)\big)^p}} &\lesssim \frac{m_n^{p/2}}{\Delta_n^{-p/2} v_n^{2p-pY/2}} \sum_{i=1}^nK^p_{m_n\Delta_n}(t_{i-1}-\tau)(\Delta_n v_n^{2p-Y}+\Delta_n^{p})\notag\\
    &\lesssim \frac{v_n^{(p/2-1)Y}}{m_n^{p/2-1}\Delta_n^{p/2-1}}\int K^p(x)dx +\frac{\Delta_n^{p/2}}{m_n^{p/2-1}v_n^{2p-pY/2}}\int K^p(x)dx.\label{th2_momentvanish}
\end{align}
In \eqref{th2_momentvanish} the first term vanishes for any $p>2$ since $m_n^{-1}\Delta_n^{-1}v_n^Y \rightarrow 0$. For the second term, note that it can be written as $\Big(m_n^{1/p}\Delta_nv_n^{-2}\Delta_n^{-1/2}m_n^{-1/2}v_n^{Y/2}\Big)^p$. By taking $p\downarrow2$, the second term vanishes given $\Delta_nv_n^{-2}\lesssim \Delta_n^{1-2\beta}\rightarrow 0$ with $\beta<\frac{1}{2}$ and $m_n^{-1}\Delta_n^{-1}v_n^Y \rightarrow 0$.

In view of \eqref{th2_momentvanish} and \eqref{th2limitvar}, what remains is to check the martingale technical condition from Theorem 2.2.15 in \cite[]{jacod2012discretization}. Specifically, due to \eqref{kernel_convergence}, it requires us to show that as \(n \to \infty\),
\[
{\sum_{i=1}^n \mathbb E_{i-1} \left[ \xi_i \Delta_n M \right]}=\sum_{i=1}^n \mathbb E_{i-1} \left[ Z_i \Delta_n M \right] \overset{P}{\to} 0,
\]
when \(M = W\) or \(M\) is a bounded martingale orthogonal to \(W\). Equivalently, it suffices to prove that
\[
\frac{m_n^{1/2}}{\Delta_n^{-1/2} v_n^{2-Y/2}} \sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau) \mathbb E_{i-1} \left[ (\Delta_i X)^2 f(\Delta_i X) \Delta_i M \right] = o_P(1),
\]
where \(f(x) := \mathbf{1}_{\{v_n < |x| \leq \zeta v_n\}}\).
Similar to the proof of Lemma 12 in \cite{BONIECE2024}, there exists a \(C^2\) smooth approximation \(f_n\) of \(f\) such that for any \(\eta > 0\),
\[
\mathbf{1}_{\{v_n(1+ \frac{2}{3}v_n^\eta) < |x| < \zeta v_n(1 - \frac{2}{3}v_n^\eta)\}} \leq f_n(x) \leq \mathbf{1}_{\{v_n(1 + \frac{1}{3}v_n^\eta) < |x| < \zeta v_n(1 - \frac{1}{3}v_n^\eta)\}},
\]
\[
|f'_n(x)| \leq \frac{C}{v_n^{1+\eta}}, \quad |f''_n(x)| \leq \frac{C}{v_n^{2+2\eta}}.
\]
Then, the result follows from the following two limits
\begin{align}\label{diff_fn_approximation}
    \frac{m_n^{1/2}}{\Delta_n^{-1/2} v_n^{2-Y/2}} \sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau) \mathbb E_{i-1} \left[ (\Delta_i X)^2 (f - f_n)(\Delta_i X) \Delta_i M \right] \to 0,
\end{align}
and
\begin{align}\label{diff_fn_technical}
    \frac{m_n^{1/2}}{\Delta_n^{-1/2} v_n^{2-Y/2}} \sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau) \mathbb E_{i-1} \left[ (\Delta_i X)^2 f_n(\Delta_i X) \Delta_i M \right] = o_P(1).
\end{align}
{While the proof of \eqref{diff_fn_approximation} is relatively straightforward, the proof of \eqref{diff_fn_technical} is  nontrivial and lengthy.
This is shown in Appendix \ref{TecnPrf2}.}




\end{proof}
Finally we are ready to formally construct our debiasing estimators.  Our debiasing steps follow the same idea as in \cite{Jacod2014}. Write $\vec \zeta_k=(\zeta_1,\dots,\zeta_k)$, where $\zeta_1,\dots,\zeta_k>1$ are fixed constants. Let
\begin{align}\label{debias_formula}
\widetilde c^{(k)}_n(m_n,v_n,\vec \zeta_k) &:= \widetilde c^{(k-1)}_n(m_n, v_n,\vec \zeta_{k-1}) \nonumber\\
&\quad- \frac{\big(\widetilde c^{(k-1)}_n(m_n,\zeta_k v_n,\vec \zeta_{k-1})-\widetilde c^{(k-1)}_n(m_n, v_n,\vec \zeta_{k-1})\big)^2}{\widetilde c^{(k-1)}_n(m_n,\zeta_k^2 v_n,\vec \zeta_{k-1})-2\widetilde c^{(k-1)}_n(m_n,\zeta_k v_n,\vec \zeta_{k-1})+\widetilde c^{(k-1)}_n(m_n, v_n,\vec \zeta_{k-1})},
\end{align}
for each $k\geq 1$, with $\widetilde c^{(0)}_n(m_n, v_n) = \hat c_n(m_n, v_n) $. Our next theorem is the second main result of this paper; it establishes a CLT for the  feasible estimator $\widetilde c^{(2)}_n(m_n,v_n,\vec \zeta_2)$.
\begin{theorem}\label{main}
     Let
    \begin{align}
    \tilde Z^{(2)}_n(m_n, v_n) &:= \sqrt{m_n}\Big(\widetilde c^{(2)}_n(m_n,v_n,\vec \zeta_2) -\hat c_{n,2}(m_n)\Big).\nonumber
\end{align}
Then, under the assumptions and notation of Theorems  \ref{clt} and \ref{thm_clt_difference}, as $n\rightarrow \infty$,
\begin{align}\label{clt_spot2}
    (\tilde Z^{(2)}_n(m_n, v_n),\tilde Z'_n(m_n))
\xrightarrow{ st} \mathcal (Z_1,{Z_2}).
\end{align}
Furthermore,
if we set $$m_n\sqrt{\Delta_n} \rightarrow \theta, \quad \text{with} \quad\theta\in{[0,\infty)},$$
we have the following stable convergence in law, as $n\rightarrow \infty$:
\begin{equation}\label{MnDebRes2}
    \sqrt{m_n} (\tilde c^{(2)}_n(m_n, v_n,\vec \zeta_2)-\sigma_\tau^2) \xrightarrow{ st} Z_1 + \theta{Z_2}.
\end{equation}
\end{theorem}


\begin{remark}\label{twostep_remark}
Recall that Remark \ref{SlyRmk1} identifies two bandwidth regimes of primary importance: (i) $0<\theta<\infty$, and (ii) $\theta=0$, both with a {convergence} rate dictated by $m_n^{-1/2}.$ In case (i),  a rate-optimal (feasible) CLT for $\tilde c_n^{(2)}(m_n, v_n,\vec \zeta_2)$ is attained in \eqref{MnDebRes2} since $m_n^{-1/2}\asymp \Delta_n^{1/4}$, provided $1\leq Y<20/11$, but no CLT is available in case (i) when $Y\geq 20/11$. In case (ii), the rate is strictly slower than $\Delta_n^{1/4}$ (see Remark \ref{SlyRmk1} for details), but a CLT still holds for $\tilde c^{(2)}_n(m_n, v_n,\vec \zeta_2)$ even when $20/11\leq Y<2$ at the expense of a deteriorating rate.  Indeed, since $m_n\ll \Delta_n^{-5}v_n^{8+Y}$ and $v_n\ll\Delta_n^{\frac{1}{4-Y}}$, we have $m_n\ll \Delta_n^{(6Y-12)/(4-Y)}$ and hence $m_n^{-1/2}\gg \Delta_n^{(6-3Y)/(4-Y)}$, which tends to zero more slowly as $Y$ gets closer to $2$.
\end{remark}


\begin{remark}
In Theorem \ref{main}, we proceeded with a two-step debiasing procedure to iteratively remove bias terms in $A(v_n,m_n)$ one at a time. However, if the second-order term in \eqref{AnTerm} is negligible when multiplied by $u_n^{-1}\sqrt{m_n}$ (and, thus, \eqref{clt_difference} is valid replacing $A(v_n,m_n)$ with only the first-order term $\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau)C_{1,i}\Delta_n v_n^{2-Y}$), then only one debiasing step is needed. Specifically, for one-step debiasing it suffices that
\[
\sqrt{m_n}\sum_{i=1}^n K_{m_n\Delta_n}(t_{i-1}-\tau)C_{1,i}\Delta_n^2 v_n^{-Y}\ll 1.
\]
Therefore, we only require that $u_n^{-1}\sqrt{m_n}\Delta_nv_n^{-Y}\ll 1$ or, equivalently, $m_n\ll \Delta_n^{-3} v_n^{Y+4}$. In the optimal-rate bandwidth regime $m_n\asymp \Delta_n^{-1/2}$, this imposes the relation $\Delta_n^{\frac{5}{2(Y+4)}}\ll v_n$, but since $v_n\ll \Delta_n^{\frac{1}{4-Y}}$, this yields the restriction $Y<12/7$. Similarly as in Remark \ref{twostep_remark}, if $Y\geq 12/7$ and $\theta=0$, the one-step debiased estimator $\tilde c^{(1)}_n(m_n, v_n,\zeta_1)$ will still enjoy a CLT centered at $\sigma_\tau^2$, but with a rate strictly slower than $\Delta_n^{\frac{4-2Y}{4-Y}}$, which tends to zero more slowly as $Y$ gets closer to $2$.
\end{remark}

\begin{proof}
Denote $\mathcal{I} = \{(1,1),(2,1)\}$ and recall that
\begin{align}\label{original}
    \tilde Z_n(m_n, v_n) = \sqrt {m_n} \left(\hat c_n(m_n, v_n) - \frac{\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2}{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}} -{A(v_n,m_n)}\right),
\end{align}
where
\begin{align*}
    A(v_n,{m_n}) = \frac{\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau){C_{1,i} \Delta_n} v_n^{2-Y}}{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}} + \frac{\sum_{i=1}^nK_{m_n\Delta_n}(t_{i-1}-\tau){D_{1,i}} {\Delta_n^2} v_n^{-Y}}{{\Delta_n{\sum_{j=1}^nK_{m_n\Delta_n}(t_{j-1}-\tau)}}}
    =: \sum_{(i,j)\in \mathcal I} a_{i,j}( v_n).
\end{align*}
We introduce the following notation:
\begin{equation}\label{INNH}
\begin{aligned}
&\eta_{i,j}(\zeta) := \zeta^{2-2(i-j)-jY}-1,\nonumber\\
    &\Psi_n:= u_n^{-1}\big(\tilde Z_n(\zeta_{1}^2 m_n, v_n) - 2\tilde Z_n(\zeta_{1} m_n, v_n) + \tilde Z_n( m_n, v_n)\big)
    = O_P(1),\nonumber\\
    &\Phi_n := u_n^{-1}\big(\tilde Z_n(\zeta m_n, v_n)-\tilde Z_n( m_n, v_n)\big) = O_P(1),
\end{aligned}
\end{equation}
where $O_P(1)$ is a consequence of Theorem \ref{thm_clt_difference}.
Note that, for any $(i,j)$ we have
\begin{equation}\label{difference}
    \begin{aligned}
    &a_{i,j}(\zeta v_n) - a_{i,j}( v_n) = \eta_{i,j}(\zeta)a_{i,j}( v_n),\\
   & a_{i,j}(\zeta^2 v_n) - 2a_{i,j}(\zeta v_n)  + a_{i,j}( v_n) = \eta_{i,j}^2(\zeta)a_{i,j}( v_n).
\end{aligned}
\end{equation}
Recall $\Delta_nv_n^{-2}\ll1$ gives the order comparison:
\begin{align}
    a_{1,1}( v_n) \gg  a_{2,1}( v_n).
\end{align}
In the first stage of debiasing, the leading order term $a_{1,1}( v_n)$ is removed.
\subsubsection*{First step: removal of $a_{1,1}( v_n)$}
Let
\begin{align}\label{debias1}
    \widetilde c_n^{(1)}(m_n,v_n,\zeta_{1}) &= \hat c_n(m_n, v_n) - \frac{\Big(\hat c_n(m_n,\zeta_{1} v_n)-\hat c_n(m_n, v_n)\Big)^2}{{\hat c_n(m_n,\zeta_{1}^2 v_n)} - 2\hat c_n(m_n,\zeta_{1} v_n) + \hat c_n(m_n, v_n)}\nonumber\\
    &=:\hat c_n(m_n, v_n) - \hat a_{1,1}( v_n).
\end{align}
Note that, due to \eqref{original}-\eqref{INNH},
\begin{align*}
\hat{c}_n(m_n,\zeta v_n)-\hat{c}_n(m_n,v_n)
&=m_n^{-1/2}\left(\tilde Z_n(m_n, \zeta v_n)-\tilde Z_n(m_n, v_n)\right)+A_n(\zeta v_n,m_n)-A_n(v_n,m_n)\\
&=m_n^{-1/2}u_n\Phi_n+A_n(\zeta v_n,m_n)-A_n(v_n,m_n).
\end{align*}
Similarly, we can write:
\begin{align*}
\hat{c}_n(m_n,\zeta^2 v_n)-2\hat{c}_n(m_n,\zeta v_n)+\hat{c}_n(m_n,v_n)
&=m_n^{-1/2}u_n\Psi_n+A_n(\zeta^2 v_n,m_n)-2A_n(\zeta v_n,m_n)+A_n(v_n,m_n).
\end{align*}
Then, in the light of Proposition \ref{prop} and \eqref{difference}, we may expand $\hat a_{1,1}( v_n)$ as follows:
\begin{equation}\label{hata11}
\begin{aligned}
    \hat a_{1,1}( v_n)
    &= \frac{\big(\sum_{\substack{(r,s)\in \mathcal I}} \eta_{r,s}a_{r,s}( v_n)+m_n^{-1/2}u_n\Phi_n\big)^2}{\sum_{\substack{(r,s)\in \mathcal I}} \eta^2_{r,s}a_{r,s}( v_n)+m_n^{-1/2}u_n\Psi_n}\\
     &=a_{1,1}( v_n) +O_P(m_n^{-1/2}u_n) \\
     &\quad+ \eta_{2,1}a_{2,1}( v_n)\frac{(2\eta_{1,1} - \eta_{2,1})a_{1,1} + \eta_{2,1}a_{2,1}( v_n)}{\eta^2_{1,1}a_{1,1}( v_n) + \eta^2_{2,1}a_{2,1}( v_n)+ m_n^{-1/2}u_n \Psi_n},
\end{aligned}
\end{equation}
where $O_P(m_n^{-1/2}u_n)$ includes the cross products between $\sum_{(r,s)\in \mathcal I }\eta_{r,s}a_{r,s}( v_n)$ and $m_n^{-1/2}u_n \Phi_n=o_P(1)$, along with $(m_n^{-1/2}u_n \Phi_n)^2$ and $m_n^{-1/2}u_n \Psi_n a_{1,1}$, all divided by $\sum_{\substack{(r,s)\in \mathcal I}} \eta^2_{r,s}a_{r,s}( v_n)+m_n^{-1/2}u_n\Psi_n$.

To simplify the last term in \eqref{hata11},
note that for any variable $N = o_p\big(a_{1,1}^2( v_n)\big)$,
\begin{align}\label{bigopapprox}
    &\frac{N}{\sum_{\substack{(r,s)\in \mathcal I}} a_{r,s}( v_n) + O_p( m_n^{-1/2}u_n)} -\frac{N}{\sum_{\substack{(r,s)\in \mathcal I}} a_{r,s}( v_n)}\nonumber\\
    &\quad= -\frac{N}{\sum_{\substack{(r,s)\in \mathcal I}} a_{r,s}( v_n)} \cdot \frac{O_P(m_n^{-1/2}u_n)}{\big(\sum_{\substack{(r,s)\in \mathcal I}} a_{r,s}( v_n) + O_P(m_n^{-1/2}u_n)\big)}\nonumber\\
    &\quad=\frac{-N}{a_{1,1}^2( v_n)(1+o_p(1))}\cdot O_P(m_n^{-1/2}u_n) \nonumber\\
    &\quad =o_P(m_n^{-1/2}u_n),
\end{align}
{where above, for simplicity, we omitted the coefficients associated with $a_{i,j}( v_n)$. The above shows that we can drop the term $m_n^{-1/2}u_n \Psi_n$ in the denominator and can write \eqref{hata11} as follows
\begin{equation*}
\begin{aligned}
    \hat a_{1,1}( v_n)
     &=a_{1,1}( v_n) +O_P(m_n^{-1/2}u_n) \\
     &\quad+ \eta_{2,1}a_{2,1}( v_n)\frac{(2\eta_{1,1} - \eta_{2,1})a_{1,1} + \eta_{2,1}a_{2,1}( v_n)}{\eta^2_{1,1}a_{1,1}( v_n) + \eta^2_{2,1}a_{2,1}( v_n)}\\
     &=a_{1,1}( v_n)+\tilde{\eta}_{2,1}a_{2,1}+\breve{\eta}_{2,1}\frac{a_{2,1}^2(v_n)}{\eta^2_{1,1}a_{1,1}( v_n) + \eta^2_{2,1}a_{2,1}( v_n)}+O_P(m_n^{-1/2}u _n),
\end{aligned}
\end{equation*}
for some constants $\tilde{\eta}_{2,1}$ and $\breve{\eta}_{2,1}$. Note that the third term above is $O_P(a_{2,1}^2/a_{1,1})$ and, thus, this term is $o_P(m_n^{-1/2}u_n)$ due to our assumption  $m_n \ll \Delta_n^{-5} v_n^{8+Y}$. We conclude that
\begin{equation}\label{hata112}
    \hat a_{1,1}(v_n)
     =a_{1,1}(v_n)+\tilde{\eta}_{2,1}a_{2,1}(v_n)
     +O_P(m_n^{-1/2}u _n).
\end{equation}
We now show the first debiasing step eliminates $a_{1,1}(v_n)$. Indeed, denoting $\tilde{a}_{2,1}(v_n):=(1-\tilde{\eta}_{2,1})a_{2,1}(v_n)$, in light of \eqref{original}, \eqref{debias1}, and \eqref{hata112} and recalling that $A(v_n,m_n)=a_{1,1}+a_{1,2}$, we see that
\begin{align*}
    \tilde Z^{(1)}_n(m_n, v_n) &:= \sqrt{m_n} \Big(\widetilde c_n^{(1)}(m_n, v_n,\zeta_{1}) - \tilde{a}_{2,1}(v_n)-\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2\Big)\nonumber\\
    &=\sqrt{m_n} \Big(\hat c_n(m_n, v_n) -\hat{a}_{1,1}(v_n)-\tilde{a}_{2,1}(v_n)-\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2\Big)\nonumber\\
    &= \tilde Z_n(m_n, v_n) + O_P(u_n).
\end{align*}
Therefore, \eqref{clt_spot} implies that}
\begin{align}\label{clt_spot2b}
    (\tilde Z_n^{(1)}(m_n, v_n),\tilde Z'_n(m_n))
\xrightarrow{ st} \mathcal (Z_1,Z_2).
\end{align}










\subsubsection*{Second step: removal of $a_{2,1}( v_n)$}
Similar to \eqref{debias1}, we can define
\begin{align}
    \widetilde c_n^{(2)}(m_n, v_n,\zeta_{2},\zeta_{1}) &= \widetilde c_n^{(1)}(m_n, v_n,\zeta_{1}) - \frac{\Big(\widetilde c_n^{(1)}(m_n,\zeta_{2} v_n,\zeta_{1})-\widetilde c_n^{(1)}(m_n, v_n,\zeta_{1})\Big)^2}{\widetilde c_n^{(1)}(m_n,\zeta_{2}^2 v_n,\zeta_{1}) - 2\widetilde c_n^{(1)}(m_n,\zeta_{2} v_n,\zeta_{1}) + \widetilde c_n^{(1)}(m_n, v_n,\zeta_{1})}\nonumber\\
    &=:\widetilde c_n^{(1)}(m_n,\zeta_{2} v_n,\zeta_{1}) - \hat a_{2,1}( v_n).
\end{align}
Similar to \eqref{hata11}, we may expand $\hat a_{2,1}( v_n)$ as follows:
\begin{align}
    &\hat a_{2,1}( v_n)
    = {\tilde{a}_{2,1}( v_n)}
    +O_P(m_n^{-1/2}u_n).
\end{align}
Then we see that
\begin{align}
    \tilde Z^{(2)}_n(m_n, v_n) &:= \sqrt{m_n} \Big(\widetilde c_n^{(2)}( m_n,v_n,\zeta_{2},\zeta_{1}) -\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2\Big)\nonumber\\
     &=\sqrt{m_n} \Big(\widetilde c_n^{(1)}( m_n,v_n,\zeta_{2},\zeta_{1}) -\hat{a}_{2,1}(v_n)-\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2\Big)\nonumber\\
     &=\sqrt{m_n} \Big(\widetilde c_n^{(1)}( m_n,v_n,\zeta_{2},\zeta_{1}) -\tilde{a}_{2,1}(v_n)-\Delta_n\sum_{i=1}^{n}K_{m_n\Delta_n}(t_{i-1}-\tau)\sigma_{t_{i-1}}^2\Big)+O_P(u_n)\nonumber\\
    &= \tilde Z^{(1)}_n(m_n, v_n) + O_P(u_n).
\end{align}
Again, \eqref{clt_spot2b} now implies that
\begin{align}
    (\tilde Z_n^{(2)}(m_n, v_n),\tilde Z'_n(m_n))
\xrightarrow{ st} \mathcal (Z_1,Z_2).
\end{align}
To conclude the last two statements of the theorem, we follow the same arguments as in Remark \ref{SlyRmk1}.
\end{proof}

\section{Simulation} \label{SectionSimul}
In the numerical experiments below, we evaluate the performance of our debiased estimators under a Heston-type
volatility model in two different settings. First, for comparison, we replicate the parameter setting and sampling scheme used in [15], which incorporates a symmetric stable jump component, although this configuration is arguably unrealistic for most financial applications (e.g., the overall annualized volatility is set to 1). Second, we consider a parameter setting that better reflects the empirical features of financial data. In this case, we also truncate the increments of the stable L\'evy process, as is commonly done in applications, and investigate the impact of different choices of the threshold parameter $v_n$.
\subsection{Simulation with Stable Jump Component}\label{SubSectSimulation}
Following \cite{LIU2018}, we consider the following model:
\begin{equation}\label{HestonModGen}
\begin{aligned}
X_t &= X_0+\int_0^t b_tdt+ \int_0^t\sqrt{V_s}\,dW_s + J_t,\\
V_t &=  V_0 + \int_0^t\kappa (\theta - V_s)ds + \xi\int_0^t \sqrt{V_s}\,dB_s,
\end{aligned}
\end{equation}
where $B$ and $B'$ are correlated standard Brownian motions with correlation $\rho$, $J$ is a strictly symmetric stable L\'evy process \footnote{The \texttt{rstable} package in \texttt{R} was used to simulate the $Y$-stable portion of the jump component in all cases.} with Blumenthal-Getoor index $Y$ and scale parameter $1$ (note \cite{LIU2018} includes an additional finite-activity component which we omit for simplicity). In this section, we take the same parameter values as in \cite{LIU2018}:
\[
    X_0=1,\quad V_0=0,\quad \rho=0,\quad b_t\equiv 1, \quad\kappa=0.03,\quad\theta=1,\quad \xi=1.5.
\]

In this simulation, we compare the performance of our estimator $\hat c(m_n,v_n)$ defined in \eqref{kernel_spot_estimator}, $\tilde c^{(1)}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$ defined in \eqref{debias_formula} to the estimators $\hat \sigma^2_{\tau,n}(u_n,h)$ and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ introduced in \cite{LIU2018}, which are based on the empirical characteristic approach of \cite{Jacod2014}. Specifically, $\hat \sigma^2_{\tau,n}(u_n,h)$\footnote{\cite{LIU2018} defines $\hat \sigma^2_{\tau,n}(u_n,h) = \bar \sigma^2_{\tau,n}(u_n,h)- \frac{2h}{u_n^2\Delta_n}(\sinh(\bar \sigma^2_{\tau,n}(u_n,h)))^2$ for simulation. The difference lies in the reciprocal placement of $h$ and $\Delta_n$ in the bias correction term. We interpret the version in \cite{LIU2018} as a likely typo and instead follow the formulation consistent with \cite{Jacod2014}.} and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ are defined as
\begin{align*}
    \hat \sigma^2_{\tau,n}(u_n,h) &=\bar \sigma^2_{\tau,n}(u_n,h)- \frac{2\Delta_n}{u_n^2h}(\sinh(\bar \sigma^2_{\tau,n}(u_n,h)))^2,\\
    \tilde \sigma^2_{\tau,n}(\lambda,u_n,h) &= \hat \sigma^2_{\tau,n}(u_n,h) - \tilde B_{\tau,n}(\lambda,u_n,h),
\end{align*}
where
\begin{align*}
\bar \sigma^2_{\tau,n}(u_n,h) &= \frac{-2}{u_n^2}\mathrm {log}\left(S_{\tau,n}(u_n,h) \vee\sqrt{\frac{\Delta_n}{h}}\right),\\
S_{\tau,n}(u_n,h) &= \Delta_n\sum_{i=1}^nK_h(i\Delta_n-\tau)\cos\left(\frac{u_n\Delta_iX}{\sqrt{\Delta_n}}\right),
\end{align*}
and $\tilde B_{\tau,n}(\lambda,u_n,h)$ is a bias-correction term\footnote{The formula {of $\tilde B_{\tau,n}(\lambda,u_n,h)$} in \cite{LIU2018} forces each summand in the  denominator to be nonpositive. However, in our experiments, we found that omitting control of the sign of these terms significantly improved the performance of the estimators. }
\begin{align*}
    \tilde B_{\tau,n}(\lambda,u_n,h) & = \frac{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\left[{\hat\sigma}^2_{(i-1)m_n\Delta_n,n}(\lambda pu_n,h) - {\hat\sigma}^2_{(i-1)m_n\Delta_n,n}(pu_n,h)\right]\cdot{\left[(\hat \sigma^2_{\tau,n}(\lambda u_n,h) -  \hat \sigma^2_{\tau,n}(u_n,h))\wedge 0\right]}}{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\big({\hat\sigma}^2_{(i-1)m_n\Delta_n,n}(\lambda^2 pu_n,h) - 2{\hat\sigma}^2_{(i-1)m_n\Delta_n,n}((\lambda pu_n,h) + {\hat\sigma}^2_{(i-1)m_n\Delta_n,n}(pu_n,h)\big)},
\end{align*}
which aggregates spot estimators across $t$ for improved finite-sample performance. This approach was also used in \cite{Jacod2014}.

For comparison, for our one- and two-step debiased estimators $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$, we also adopt the debiasing adjustments of \cite{LIU2018} and \cite{Jacod2014} described above. Specifically, denoting $\tilde{c}^{(k)}_{n,\tau}(m_n,v_n,\vec \zeta_k)$ the estimator $\tilde{c}^{(k)}_{n}(m_n,v_n,\vec \zeta_k)$ at time $\tau$, we consider
\begin{align*}
    \widetilde c^{(k)}_{n,\tau}(m_n,v_n,\vec \zeta_k) &:= \widetilde c^{(k-1)}_{n,\tau}(m_n, v_n,\vec \zeta_{k-1}) \nonumber\\
&\quad- \big(A^{(k)}_{n}(m_n,v_n,\vec \zeta_k)\vee0\big)\big(\widetilde c^{(k-1)}_{n,\tau}(m_n,\zeta_k v_n,\vec \zeta_{k-1})-\widetilde c^{(k-1)}_{n,\tau}(m_n, v_n,\vec \zeta_{k-1})\big)\vee 0,
\end{align*}
where $A^{(k)}_{{n}}:=A^{(k)}_{n}(m_n,v_n,\vec \zeta_k)$ is given by
$$A_n^{(k)}
=\pm\frac{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\big(\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_kp_k v_n,\vec \zeta_{k-1})-\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n, p_kv_n,\vec \zeta_{k-1})\big)}{\sum_{i=1}^{\lfloor\frac{1}{m_n\Delta_n}\rfloor}\big(\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_k^2 p_kv_n,\vec \zeta_{k-1})-2\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,\zeta_k p_kv_n,\vec \zeta_{k-1})+\widetilde c^{(k-1)}_{n,im_n\Delta_n}(m_n,p_k v_n,\vec \zeta_{k-1})\big)},$$
where the sign of the bias terms are chosen as in \cite{BONIECE2024} and $c^{(0)}_{n,\tau}(m_n,v_n,\vec \zeta_0) = \hat c(m_n,v_n)$ at $\tau$, i.e., $A_n^{(1)}$ is positive and $A_n^{(2)}$ is negative, so as to ensure compatibility with the theoretical sign of the terms in $A(v_n,m_n)$.

To evaluate estimation performance, we simulate $M$ independent paths and aggregate various pathwise error measures computed along each simulated path for each of the estimators considered above.  Denoting the true value of the spot variance at $t_i$ in {the} $j$-th path as $\sigma^2_{t_i,j}:=V_{t_i,j}$ and a given  estimator as $\hat \sigma^2_{t_i,j}$, we consider an estimate of  the pathwise root mean squared error (RMSE) on the time grid $\{l_i = i\floor {n/100}\}_{i=10,\dots,90}$ as follows:
$$\widehat{RMSE} := \sqrt{\frac{1}{M}\sum_{{j=1}}^M MSE_j}\,,$$
where $MSE_j := \frac{1}{81} \sum_{i=10}^{90}(\hat \sigma^2_{l_i,j}-\sigma^2_{l_i,j})^2$. Similarly, we also consider the mean absolute relative error $\widehat{ARE}$ and the mean relative error $\widehat{RE}$:
$$\widehat{ARE} := \frac{1}{M}\sum_{j=1}^M AE_j \;\text{ and }\;\widehat{RE} := \frac{1}{M}\sum_{j=1}^M E_j,$$
where $AE_j := \frac{1}{81} \sum_{i=10}^{90}|\hat \sigma^2_{l_i,j}-\sigma^2_{l_i,j}|/\sigma^2_{l_i,j}$ and $E_j := \frac{1}{81} \sum_{i=10}^{90}(\hat\sigma^2_{l_i,j}-\sigma^2_{l_i,j})/\sigma^2_{l_i,j}$.

For our estimators $\hat c_n(m_n,\Delta_n)$, $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$, we use the exponential kernel $K(x) = \exp({-|x|})/2$ as well as the two-sided uniform kernel $G(x) = \mathbf{1}_{-1\leq |x|\leq 1}/2$, and set $m_n = \Delta_n^{-1/2}$, $v_n = \sqrt{BV} \Delta_n^{5/12}$, where $BV = \frac{\pi}{2}\sum_{i=2}^n|\Delta_iX||\Delta_{i-1}X|/T$ is the bipower estimator for the integrated volatility.
For the estimators $\hat \sigma^2_{\tau,n}(u_n,h)$ and $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$ of \cite{LIU2018}, we take the kernel $K_3(x) = \frac{15}{16}(1-x^2)^2\mathbf{1}_{|x|\leq 1}$ as specified on \cite[p.~1964]{LIU2018}\footnote{$K_3$ is the best-performing kernel among those considered in \cite{LIU2018} that satisfies the assumptions of their results (in particular, that is continuously differentiable as assumed by their main theorem).}  We also use the best-performing bandwidth as reported in their simulations, $h=\Delta_n^{0.51}$ and $u_n = \frac{\Delta_n^{0.0025}}{\sqrt {BV}}$\footnote{Instead of using bipower variation for \( u_n \), the authors of \cite{LIU2018} employed a scaled local bipower variation defined as
$BV_\tau = \frac{\pi}{2(i_2 - i_1)\Delta_n} \sum_{i = i_1}^{i_2} |\Delta_i X||\Delta_{i+1} X|,$ where $ i_1 = \left\lfloor\frac{\tau - h}{\Delta_n} + 1\right\rfloor \vee 1 $ and $ i_2 = \left\lfloor\frac{\tau + h}{\Delta_n} + 1\right\rfloor \wedge n $. However, in our experiments, this choice lead to large RMSE when aggregating errors across various \( \tau \), particularly for \( Y = 1.2 \). In contrast, the estimators demonstrated greater stability with our proposed choice of \( u_n \).}. The remaining tuning parameters $\vec\zeta_1$, $\vec\zeta_2$, $\lambda$, $p$, $p_1$, $(p_1,p_2)$ are selected via a grid search that minimizes $\widehat {RMSE}$\footnote{Specifically, to each point in the grid, we simulated $M=100$ independent paths to estimate the $\widehat{RMSE}$ for the associated parameter settings.}. The grid for $\zeta$ and $\lambda$ ranged from 1.1 to 1.9 with increments of 0.05, while the grid for $p$ spanned from 0.1 to 0.9 with increments of 0.05. For reproducibility, where applicable, we also report the auxiliary tuning parameter values found in our grid search for each considered estimator.

As in \cite{LIU2018}, we take $T=1$ month and $n = 8580$, roughly corresponding to a 5-minute sampling frequency (assuming 24 hours of  trading). For additional comparison, we also include the results for $n = 42900$,  corresponding to roughly 1-minute frequency data in this setup. Using $M = 1000$ iterations, we report the results in Tables \ref{Table1} and \ref{Table2}.
We consider the values $Y\in\{0.8,1.2,1.6,1.75\}$, which correspond to the cases where, in principle, no debiasing is needed ($Y\in\{0.8,1.2\}$), one step of debiasing is needed ($Y=1.6$), and two steps are needed $(Y=1.75)$ to attain asymptotic efficiency {according to our  theory, though we may observe very different finite sample behavior}.

As shown in Tables \ref{Table1} and \ref{Table2},  the two-step debiasing estimator $\tilde{c}_n^{(2)}$ consistently delivers the best performance, with the lowest RMSE and ARE, and generally well-controlled error.  The advantage of two-step debiasing becomes especially pronounced as $Y$ increases:  While the performance of $\hat \sigma^2_n$ and $\widetilde \sigma^2_n$ deteriorates sharply, the one- and two-step debiased estimators exhibit markedly greater robustness against increased jump activity.  Notably, two-step debiasing yields improvement even where it is not strictly needed for asymptotically efficient estimation ($Y=0.8,1.2$). The pattern is similar at both 5-minute and 1-minute frequencies, where the two-step debiasing method is the only one that remains stable across all considered settings, offering reductions in RMSE often by a factor of 2 to 5 relative to the considered benchmarks.

Across kernel choices for $\tilde c_n^{(i)}$, the exponential kernel systematically outperforms the uniform kernel across all values of $Y$ and at both 1- and 5-minute sampling frequencies. This finding is consistent with the established theoretical optimality of exponential kernels in related settings where debiasing is not required \cite{Figueroa-Lopez_Wu_2024}, and more broadly underscores the advantages of using unbounded kernels in spot volatility estimation. Overall, the two-step estimator with the exponential kernel achieves the strongest and most reliable performance among all configurations considered.








\begin{table}[htbp]

\centering
\caption{Estimation performance at a 5-minute sampling frequency, based on a set of $M=1000$ paths. The simulated data were generated at over a $T=1$ month horizon using the model setup of \cite{LIU2018}. The indicated tuning parameters are from the grid search and correspond to the values of $\lambda,p$ for $\tilde \sigma^2_{\tau,n}$ and to the values of $\vec \zeta,\vec p$ for $c_{n,\tau}^{(i)}$, $i=1,2.$}\label{Table1}
\scalebox{0.87}{
\begin{tabular}{|c|c|c|c|>{\centering\arraybackslash}p{3cm}|c|c|c|>{\centering\arraybackslash}p{3cm}|}
\hline
 &\multicolumn{4}{|c|}{$Y=0.8$} & \multicolumn{4}{|c|}{$Y=1.2$} \\
\hline
\textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} &\textbf{RE}&\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}
& \textbf{RMSE} & \textbf{ARE} &\textbf{RE} &\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}} \\
\hline
 $\hat \sigma^2_{\tau,n}(u_n,h)$       & 0.2024 & 0.1074 &\textbf{\smash{\scalebox{0.96}[1]{0.0070}}} &-& 0.2365 & 0.1168 &0.0451 &-\\
 $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$  & 0.2355 &0.1109 &0.0094 &$\lambda = 1.6,\: p = 0.9$& 0.2288 &0.121 &-\textbf{\smash{\scalebox{0.96}[1]{0.0030}}}&$\lambda = 1.25,\: p = 0.2$\\
 $\hat c_{n,\tau,\text{exp}}(m_n,v_n)$ &0.2805 & 0.1738 &-0.1704&- &0.2509 & 0.1539 &-0.1519 &-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_1)$          & 0.2806 & 0.1740 &-0.1704 &$\zeta = 1.8,\:p =0.6$ & 0.2513 & 0.1540 &-0.1522 &$ \zeta = 1.7,\: p =0.85$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_2)$         & \textbf{\smash{\scalebox{0.96}[1]{0.1215}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0608}}} &0.0099 &$\vec \zeta = (1.7,1.8)$,\quad$\vec p =(0.5,0.85)$ & \textbf{\smash{\scalebox{0.96}[1]{0.1421}}} &  \textbf{\smash{\scalebox{0.96}[1]{0.0740}}} &0.0234 &$\vec \zeta = (1.9,1.25)$,\quad$\vec p =(0.5,0.65)$\\
 $\hat c_{n,\tau,\text{unif}}(m_n,v_n)$&0.2932 & 0.1768 &-0.1722&-&0.2634 & 0.1568 &-0.1527 &-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_1)$           & 0.2934 & 0.1769 &-0.1726  & $\zeta = 1.85,\:p =0.85$&0.2635 & 0.1568 &-0.1528 &$  \zeta = 1.65,\: p =0.8$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_2)$         & 0.1824 & 0.0861 &0.0106  &$\vec \zeta = (1.6,1.9)$,\quad$\vec p =(0.5,0.9)$&0.1983 & 0.1046 &0.0631 &$\vec \zeta = (1.9,1.25)$,\quad$\vec p =(0.6,0.15)$\\
\hline
\hline
 &\multicolumn{4}{|c|}{$Y=1.6$} & \multicolumn{4}{|c|}{$Y=1.75$} \\
\hline
\textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} &\textbf{RE}&\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}
& \textbf{RMSE} & \textbf{ARE} &\textbf{RE} &\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}\\
\hline
 $\hat \sigma^2_{\tau,n}(u_n,h)$       &  0.4551 & 0.2604 & 0.2534 &-& 0.7784 &0.4830 &0.4827&-\\
 $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$   & 0.3264 & 0.1760 & 0.1013 &$\lambda = 1.9,\: p = 0.1$& 0.5361 & 0.2984 &0.2590&$\lambda = 1.9,\: p = 0.25$\\
 $\hat c_{n,\tau,\text{exp}}(m_n,v_n)$ & \textbf{\smash{\scalebox{0.96}[1]{0.1121}}} &\textbf{\smash{\scalebox{0.96}[1]{0.0595}}}& -0.0036 &-& 0.2785 & 0.1702 &0.1688&-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_1)$          & \textbf{\smash{\scalebox{0.96}[1]{0.1121}}} &\textbf{\smash{\scalebox{0.96}[1]{0.0595}}} & -0.0036&$ \zeta = 1.75,\:  p = 0.75$ &0.1783 & 0.0978 &0.0706&$ \zeta =1.65,\: p =0.1$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_2)$         &  \textbf{\smash{\scalebox{0.96}[1]{0.1121}}} &\textbf{\smash{\scalebox{0.96}[1]{0.0595}}} & -0.0036 &$\vec \zeta = (1.9,1.75)$,\quad$\vec p =(0.7,0.25)$& \textbf{\smash{\scalebox{0.96}[1]{0.1640}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0885}}} &\textbf{\smash{\scalebox{0.96}[1]{0.0412}}}&$\vec \zeta = (1.8,1.55)$,\quad$\vec p =(0.2,0.35)$\\
 $\hat c_{n,\tau,\text{unif}}(m_n,v_n)$ & 0.1490 &0.0796 & -0.0012 &-& 0.2992 & 0.1749 &0.1684&-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_1)$          &0.1490 & 0.0796 & -0.0012&$ \zeta = 1.9,\: p =0.65$ &0.2265 & 0.1240 &0.0892&$ \zeta = 1.8,\: p =0.1$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_2)$         & 0.1490 & 0.0796 &\textbf{\smash{\scalebox{0.96}[1]{-0.0011}}} &$\vec \zeta = (1.9,1.9)$,\quad$\vec p =(0.6,0.15)$&0.2323 & 0.1279 &0.0983&$\vec \zeta = (1.9,1.9)$,\quad$\vec p =(0.1,0.2)$\\
\hline
\end{tabular}
}
\end{table}

\begin{table}[htbp]
\centering
\caption{Estimation performance at 1-minute sampling frequency, based on a set of $M=1000$ paths. The simulated data were generated at over a $T=1$ month horizon using the model setup of \cite{LIU2018}. The indicated tuning parameters are from the grid search and correspond to the values of  $\lambda,p$ for $\tilde \sigma^2_{\tau,n}$ and to the values of $\vec \zeta,\vec p$ for $c_{n,\tau}^{(i)}$, $i=1,2.$ }\label{Table2}
\centering
\scalebox{0.87}{
\begin{tabular}{|c|c|c|c|>{\centering\arraybackslash}p{3cm}|c|c|c|>{\centering\arraybackslash}p{3cm}|}
\hline
 &\multicolumn{4}{|c|}{$Y=0.8$} & \multicolumn{4}{|c|}{$Y=1.2$} \\
\hline
\textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} &\textbf{RE}&\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}
& \textbf{RMSE} & \textbf{ARE} &\textbf{RE} &\textbf{\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}} \\
\hline
 $\hat \sigma^2_{\tau,n}(u_n,h)$       & 0.1373 & 0.0725 & \textbf{\smash{\scalebox{0.96}[1]{0.0028}}} &-& 0.1434 & 0.0754 & \textbf{\smash{\scalebox{0.96}[1]{0.0222}}} &-\\
 $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$  &0.1384 & 0.0729 & 0.0043&$\lambda = 1.85,\:p =0.8$&0.1441 & 0.0758 & 0.0242&$\lambda = 1.85,\:p =0.9$\\
 $\hat c_{n,\tau,\text{exp}}(m_n,v_n)$ &0.1682 & 0.1016 &-0.0998&- &0.1494 & 0.0888 & -0.0873 &-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_1)$          & 0.1684 & 0.1016 & -0.0990 &$\zeta = 1.55,\: p =0.9$ & 0.1495 & 0.0888 &  -0.0874 &$ \zeta = 1.7,\: p =0.8$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_2)$         &  \textbf{\smash{\scalebox{0.96}[1]{0.0760}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0401}}} & 0.0050 &$\vec \zeta = (1.7,1.9)$,\quad$\vec p =(0.5,0.75)$ & \textbf{\smash{\scalebox{0.96}[1]{0.0859}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0455}}} & 0.0251 &$\vec \zeta = (1.9,1.25)$,\quad$\vec p = (0.6,0.5)$\\
 $\hat c_{n,\tau,\text{unif}}(m_n,v_n)$& 0.1788 & 0.1043 &-0.1002&-&0.1610 & 0.0920 &-0.0871 &-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_1)$           & 0.1792 & 0.1044 &-0.1002  &$
  \zeta = 1.65,\: p = 0.8$ &0.1610 & 0.0920 &-0.0871 &$  \zeta = 1.65,\: p = 0.75$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_2)$        & 0.1060 & 0.0567 &0.0050  &$\vec \zeta = (1.7,1.9)$,\quad$\vec p = (0.5,0.75)$&0.1184 & 0.0625 &0.0293 &$\vec \zeta = (1.9,1.75)$,\quad$\vec p =(0.4,0.8)$\\
\hline
\hline
 &\multicolumn{4}{|c|}{$Y=1.6$} & \multicolumn{4}{|c|}{$Y=1.75$} \\
\hline
\textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} &\textbf{RE}&\textbf{\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}}
& \textbf{RMSE} & \textbf{ARE} &\textbf{RE} &\textbf{\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}}\\
\hline
 $\hat \sigma^2_{\tau,n}(u_n,h)$        & 0.3157& 0.1853 & 0.1823 &-&0.6148 &0.3959 &0.3950&-\\
 $\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$   & 0.2456 & 0.1324 & 0.0910 &$\lambda = 1.85,\:p =0.1$ & 0.5286 &0.3575 &0.0907&$\lambda = 1.85,\:p =0.55$\\
 $\hat c_{n,\tau,\text{exp}}(m_n,v_n)$ & 0.0965 & 0.0533 & 0.0406 &-& 0.3364 & 0.2209 &0.2209&-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_1)$          & \textbf{\smash{\scalebox{0.96}[1]{0.0823}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0440}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0036}}}&$ \zeta = 1.9,\: p =0.1$ & 0.1563 & 0.0882 &0.0703&$ \zeta = 1.75,\: p =0.25$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{exp}}(m_n,v_n,\vec \zeta_2)$         & \textbf{\smash{\scalebox{0.96}[1]{0.0823}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0440}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0036}}} &$\vec \zeta = (1.9,1.65)$,\quad$\vec p =(0.1,0.3)$& \textbf{\smash{\scalebox{0.96}[1]{0.1423}}} & \textbf{\smash{\scalebox{0.96}[1]{0.0776}}} &\textbf{\smash{\scalebox{0.96}[1]{0.0253}}}&$\vec \zeta = (1.4,1.9)$,\quad$\vec p =(0.3,0.2)$\\
 $\hat c_{n,\tau,\text{unif}}(m_n,v_n)$ &0.1198 & 0.0652 &0.0415 &-&0.3464 & 0.2208 &0.2207&-\\
 $\tilde{c}^{(1)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_1)$          &0.1132 & 0.0605 &-0.0053&$ \zeta = 1.7,\: p =0.1$ &0.2076 & 0.1168 &0.0967&$ \zeta = 1.4,\: p =0.2$\\
 $\tilde{c}^{(2)}_{n,\tau,\text{unif}}(m_n,v_n,\vec \zeta_2)$         &0.1102 &  0.0591 & 0.0044 &$\vec \zeta = (1.9,1.65)$,\quad$\vec p =(0.1,0.3)$&0.2181 & 0.1243 &0.1108&$\vec \zeta = (1.5,1.85)$,\quad$\vec p =(0.2,0.2)$\\
\hline
\end{tabular}
}
\end{table}









\subsection{Simulation with truncated stable jump component}
In this section, we consider a parameter setting that more closely  reflects the empirical features of financial data. We also take a truncated stable L\'evy process as the jump component, which is common in applications.
We also study the effect of the truncation level $v_n$.

For this experiment, we fix a time unit of 1 year (i.e., $t$ is measured in years). The increments of the observed process  (corresponding to an asset's log return over $(t_{i-1},t_i]$) are given by
\begin{align}
\Delta_i X &=X_{t_i}-X_{t_{i-1}}= \int_{t_{i-1}}^{t_i}\sqrt{V_s}\,dW_s + (\Delta_i J \wedge 0.005),\,
\end{align}
where $J$ is a symmetric $Y$-stable L\'evy process with scale parameter $0.5$ and $V$ follows the same Heston model \eqref{HestonModGen} with parameter values as in \cite{BONIECE2024}:
$$\rho = -0.5,\quad \kappa = 5,\quad\xi=0.5,\quad\theta=0.16.$$
In particular, we note the annualized average volatility is $\sqrt{.16}=.4$, which is more realistic for financial data. Assuming 252 trading days per year, and a 6.5 hour trading day, we set $\Delta_n = (252*6.5*12)^{-1}$, corresponding to a sampling frequency of  5 minutes. We consider a time horizon of 3 months  ($T = \frac{1}{4}$), yielding a sample size of $ n = T/\Delta_n=4914$. In our experiments, we examine performance in the cases $Y=1.6$ and $Y=1.75$, corresponding to 1 and 2 steps of debiasing required for efficiency, and use an exponential kernel for our estimators $\hat c_n(m_n,\Delta_n)$, $\tilde c^{(1)}_{m,\tau}(m_n,v_n,\vec\zeta_1)$ and $\tilde c^{(2)}_{n,\tau}(m_n,v_n,\vec\zeta_2)$.   The tuning parameters are similar to the  previous section, with $\vec\zeta_1$, $\vec\zeta_2$, $\lambda$, $p$, $p_1$, and $(p_1,p_2)$ being selected via a grid search that minimizes $\widehat{RMSE}$ over 100 iterations (using the same grid as described in previous section), but here we take $v_n = \sqrt{BV}v_0$, for different $v_0$ to assess the sensitivity of our methods to the threshold choice. We report the performance for our estimators across various threshold choices $v_0$  in Table \ref{Table3} $(Y=1.6)$ and in Table \ref{Table4} $(Y=1.75)$ as measured by $\widehat{RMSE}$, $\widehat{ARE}$ and $\widehat{RE}$ using $M=1000$ paths. In each table, we also include the performance of the estimators of \cite{LIU2018} for comparison. We also report the auxiliary tuning parameter values found in our grid search.


As shown in Tables \ref{Table3} and \ref{Table4}, for both $Y=1.6$ and $Y=1.75$ the two-step estimator $\tilde c_n^{(2)}$ again provides the most reliable performance, delivering the lowest or near-lowest RMSE and ARE across all $v_0$ considered.    The advantage of debiasing is most pronounced at the moderate threshold choices $v_0=\Delta_n^{20/48}=\Delta_n^{5/12}$ and $v_0=\Delta_n^{21/48}$, where the two-step estimators offer reductions in RMSE over $\hat \sigma^2_n$ and $\tilde \sigma^2_n$ by roughly a factor of 2, and these improvements are observed consistently for both $Y=1.6$ and $Y=1.75$.  Remarkably, the choice $v_0=\Delta_n^{5/12}$ is the same as suggested by the theory for two-step debiasing in the integrated volatility case (see \cite{BONIECE2024}). Broadly, these findings highlight the benefit of two-step debiasing beyond specific threshold choices, offering reliable nontrivial reductions in MSE across a variety of jump activity levels.











\begin{table}[htbp]
\centering
\caption{Estimation performance across $v_0$ in a realistic parameter setting with $Y=1.6$. Results based on  $M=1000$ paths at a 5-minute sampling frequency using the exponential kernel. The indicated tuning parameters are from the grid search and correspond to the values of  $\lambda,p$ for $\tilde \sigma^2_{\tau,n}$ and to the values of $\vec \zeta,\vec p$ for $c_{n,\tau}^{(i)}$, $i=1,2.$}\label{Table3}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Threshold} & \textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} & \textbf{RE} &\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}\\
\hline
\multirow{3}{*}{$v_0 = \Delta_n^{19/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.03296 & 0.22079 &0.21617 & -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$         &  0.0291 & 0.18878 &0.16621& $\zeta=1.4,\:p=0.2$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          & 0.0295 & 0.1789 &0.12536& $\vec\zeta=(1.4,1.9),\:\vec p=(0.3,0.2)$\\
\hline
\multirow{3}{*}{$v_0 = \Delta_n^{20/48}$}

& $\hat c_{n,\tau}(m_n,v_n)$ & \textbf{\smash{\scalebox{0.96}[1]{0.02342}}} &\textbf{\smash{\scalebox{0.96}[1]{0.14157}}} &\textbf{\smash{\scalebox{0.96}[1]{0.08644}}}& -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$         & \textbf{\smash{\scalebox{0.96}[1]{0.02342}}} & \textbf{\smash{\scalebox{0.96}[1]{0.14157}}} & \textbf{\smash{\scalebox{0.96}[1]{0.08644}}}& $\zeta=1.9,\:p=0.9$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          & 0.02408  & 0.15699 &0.13156& $\vec\zeta=(1.9,1.7),\:\vec p=(0.8,0.2)$\\
\hline
\multirow{3}{*}{$v_0 = \Delta_n^{21/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.03956& 0.17759 &-0.12086& -\\
&$c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$         &   0.03956 &0.17759 &-0.12086& $\zeta=1.7,\:p=0.9$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$           & 0.03956  & 0.17759 &-0.12086& $\vec\zeta(1.5,1.2),\:\vec p=(0.9,0.4)$\\
\hline
\multirow{3}{*}{$v_0 = \Delta_n^{22/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.07364 & 0.37008 &-0.36356& -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$        & 0.07364 & 0.37008 &-0.36356& $\zeta=1.8,\:p=0.8$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          & 0.05129  & 0.34341 &0.34314& $\vec\zeta=(1.8,1.9),\:\vec p=(0.7,0.9)$\\
\hline
\multirow{2}{*}{-}
& $\hat \sigma^2_{\tau,n}(u_n,h)$  &  0.04669 & 0.28256   &0.27899& -\\
&$\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$      &0.04187  &0.2466 &0.22257& $\lambda=1.9,\:p=0.35$\\
\hline
\end{tabular}
\end{table}

\begin{table}[htbp]
\centering
\caption{Estimation performance across $v_0$ in a realistic parameter setting with $Y=1.75$. Results based on  $M=1000$ paths at a 5-minute sampling frequency. The indicated tuning parameters are from the grid search and correspond to the values of  $\lambda,p$ for $\tilde \sigma^2_{\tau,n}$ and to the values of $\vec \zeta,\vec p$ for $c_{n,\tau}^{(i)}$, $i=1,2.$} \label{Table4}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Threshold} & \textbf{Estimator} & \textbf{RMSE} & \textbf{ARE} & \textbf{RE} &\textbf{\smash{\scalebox{0.96}[1]{Tuning Param.}}}\\
\hline
\multirow{4}{*}{$v_0 = \Delta_n^{19/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.05626 & 0.38748 &0.3872 & -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$        & 0.04713 & 0.31793 &0.30345& $\zeta=1.5,\:p=0.3$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          & 0.04713 & 0.31793 &0.30345& $\vec\zeta=(1.5,1.9),\:\vec p=(0.3,0.2)$\\
\hline
\multirow{4}{*}{$v_0 = \Delta_n^{20/48}$}

& $\hat c_{n,\tau}(m_n,v_n)$ & 0.03883 & 0.26904 &0.25687& -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$        & 0.03826 & 0.21271 &0.08458& $\zeta=1.7,p=0.3$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          & 0.03464  & 0.22508 &0.0588&  $\vec\zeta=(1.7,1.9),\:\vec p=(0.2,0.2)$\\
\hline
\multirow{5}{*}{$v_0 = \Delta_n^{21/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.03203& 0.16966 &0.02212& -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$        &   0.03203 &0.16966 &0.02212& $\zeta=1.6,\:p=0.9$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$          &\textbf{\smash{\scalebox{0.96}[1]{0.02906}}}  & \textbf{\smash{\scalebox{0.96}[1]{0.16739}}} &0.06315&  $\vec\zeta=(1.7,1.9),\:\vec p=(0.8,0.2)$\\
\hline
\multirow{5}{*}{$v_0 = \Delta_n^{22/48}$}
& $\hat c_{n,\tau}(m_n,v_n)$ & 0.06220 & 0.28966 &-0.2564& -\\
& $c^{(1)}_{n,\tau}(m_n,v_n,\vec \zeta_1)$         & 0.06220 & 0.28966 &-0.2564& $\zeta=1.8,\:p=0.8$\\
& $c^{(2)}_{n,\tau}(m_n,v_n,\vec \zeta_2)$         & 0.03363  & 0.1677 &\textbf{\smash{\scalebox{0.96}[1]{-0.00909}}}&  $\vec\zeta=(1.9,1.2),\:\vec p = (0.9,0.6)$\\
\hline
\multirow{2}{*}{-}
& $\hat \sigma^2_{\tau,n}(u_n,h)$   & 0.07123 & 0.46975 &0.43293& -\\
&$\tilde \sigma^2_{\tau,n}(\lambda,u_n,h)$      &0.06628  & 0.46931 &0.42743& $\lambda=1.9,\:p=0.1$\\
\hline
\end{tabular}
\end{table}