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Fast Online Changepoint Detection
\address{Department of Economics and Management, University of Pavia, 27100
Pavia, Italy, email:[email removed]}
\address{Department of Economics and Management, University of Pavia, 27100
Pavia, Italy, email:[email removed]}
\address{University of Leicester Business School, University Road, Leicester
LE1 7RH, UK, and Department of Economics and Management, University of
Pavia, 27100 Pavia, Italy, email: [email removed]}
abstractWe study online changepoint detection in the context of a linear
regression model. We propose a class of heavily weighted statistics based on
the CUSUM process of the regression residuals, which are specifically
designed to ensure timely detection of breaks occurring early on during the
monitoring horizon. We subsequently propose a class of composite statistics,
constructed using different weighing schemes; the decision rule to mark a
changepoint is based on the largest statistic across the various
weights, thus effectively working like a veto-based voting mechanism, which
ensures fast detection irrespective of the location of the changepoint. Our
theory is derived under a very general form of weak dependence, thus being
able to apply our tests to virtually all time series encountered in
economics, medicine, and other applied sciences. Monte Carlo simulations
show that our methodologies are able to control the procedure-wise Type I
Error, and have short detection delays in the presence of breaks.
\subjclass[2020]{60F17}
\doublespacing
Introduction
In this paper, we consider the linear regression model
equation[equation omitted — 112 chars of source]
and test whether the slopes ${\Greekmath 010C} _{t}$ are constant over time - i.e. $
{\Greekmath 010C} _{t}={\Greekmath 010C} _{0}$ for all $t\geq 1$ - or not. In particular, we study
sequential/online changepoint detection: after observing ((ref)) over a training period $1\leq t\leq m$ with constant slopes $
{\Greekmath 010C} _{0}$, we check whether, as new data come in, ${\Greekmath 010C} _{t}$ differs
from the previous ${\Greekmath 010C} _{0}$ for some $t>m$. Tests are carried out in real
time, at each $t>m$, as soon as the new datapoint $\left( y_{t},\mathbf{x}
_{t}^{\prime }\right) ^{\prime }$ has been recorded.
Since the seminal contribution by page1954continuous, online
changepoint detection has proven highly relevant to all applied sciences,
and we refer to a recent review by aue2023state for the current
state of the art. In manufacturing, timely detection of malfunctioning in a
production line is important for quality control (see e.g.
page1955control, and page1955test); in medicine,
changepoint analysis is relevant in the context of sequential medical trials
(armitage1960sequential), or to detect the onset of an epileptic
seizure using EEG trajectories (see e.g. ombao); and, in
epidemiology, detecting the onset of a pandemic by monitoring whether the
number of (daily) cases starts exhibiting an explosive dynamics (see e.g.
HT2023) is crucial to public health decisions. In economics and
finance, the instability of model parameters is an issue of pivotal
importance due to its consequences on forecasting and policy decision
making; see, inter alia, the contributions by pastor2001equity, pettenuzzo2011predictability, and smith2021break. We also refer to romano2023fast, and the
references therein, for further examples illustrating the importance of
quick online changepoint detection. Finding instability in ${\Greekmath 010C} _{t}$ has
obvious implications on the reliability of a model like ((ref)),
and on decisions based on it: as hansen2001new puts it, in the
presence of a changepoint \textquotedblleft inferences about economic
relationships can go astray, forecasts can be inaccurate, and policy
recommendations can be misleading or worse\textquotedblright\ (p. 127).
The literature has developed numerous tests for the ex-post detection of
changepoints, and we refer to the reviews by horvath2014extensions
and casini2019structural, for an insightful analysis of the state of
the art and of future directions. Despite its practical importance, the
issue of real-time detection is somewhat underdeveloped compared to ex-post
testing. In a seminal contribution, chu1996monitoring develop a
methodology to monitor ((ref)), based on the CUSUM process of the
residuals. The intuition is that, given an estimate of ${\Greekmath 010C} _{0}$ (say $
\widetilde{{\Greekmath 010C} }$) computed during a training sample in which no breaks
were detected, the \textquotedblleft prediction errors\textquotedblright\ $
y_{t}-\mathbf{x}_{t}^{\prime }\widetilde{{\Greekmath 010C} }$ calculated for $t>m$ will
fluctuate around zero under the null of no changepoint, whereas they will
have a bias in the presence of a changepoint. Thus, the corresponding CUSUM
process will either fluctuate around zero with increasing variance as $t$
increases, or it will have a drift after a changepoint. Hence, if, at each $
t $, the CUSUM\ process stays within a boundary, the null of no break will
not be rejected; conversely, a break will be flagged at the first crossing
of such a boundary. chu1996monitoring derive the weak limit of the
CUSUM process. This allows to compute critical values, and ultimately to
control the procedure-wise Type I Error. In addition to ensuring size
control, the timely detection of a changepoint is also very important;
horvath2004monitoring and horvath2007sequential propose a
family of boundary functions which afford a faster break detection when a
changepoint is present. These boundary functions, which represent a major
simplification of the ones studied in chu1996monitoring, are based
on a parameterisation which depends on a user-chosen quantity (say, $0\leq
{\Greekmath 0111} \leq 1/2$) which determines the weights assigned to the CUSUM
fluctuations: as ${\Greekmath 0111} $ increases, the weight also increases, and therefore
higher power/faster detection may be expected. Several recent contributions
also propose statistics based on the CUSUM process - as a leading example,
which is drawing more and more attention in the literature,
kirch2018modified, yu2020note, romano2023fast, and
HT2023, study a standardised version of the CUSUM process (called
the \textquotedblleft Page-CUSUM\textquotedblright ) based on comparing the
average during the training period with the average computed during the
monitoring period using a moving window, and choosing the length of such a
window in order to maximise the discrepancy between the two averages in
order to ensure an \textquotedblleft optimal\textquotedblright\ detection
delay when a changepoint is present. Whilst different from our approach,
even in this case the CUSUM process is suitably weighted in order to enance
fast changepoint detection.
Hypotheses of interest and the main contributions of this paper
As mentioned above, we study sequential detection of changepoints in ((ref)), allowing for the presence of exogenous and dynamic
regressors, and for a very general form of serial dependence in both $
\mathbf{x}_{t}$ and ${\Greekmath 010F} _{t}$. After observing the model over a
training sample of size $m$ where no changes in the regression coefficients
have occurred, we test for the null hypothesis of no change occurring at
each point in time, i.e.
equation[equation omitted — 167 chars of source]
We make two main contributions. Firstly, we design a class of heavily
weighted CUSUM-type statistics which ensure fast detection for early
occurring changepoints. We call these R\'{e}nyi statistics,
inspired by horvath2020new who, building on an idea by
renyi1953theory, propose a similar idea for the purpose of offline
changepoint detection. We develop the full-blown asymptotics under the null,
and study the consistency, and the limiting behaviour of the detection
delay, under the alternative. We show that R\'{e}nyi statistics deliver an
excellent performance in the presence of early occurring breaks. On the
other hand, when breaks occur at a later stage, heavily weighted statistics
offer a worse performance compared to CUSUM-type statistics using lighter
weights. Indeed, no weighing scheme can ensure uniformly fast changepoint
detection, and different values of ${\Greekmath 0111} $ yield faster detection for
different break locations (see also kirch2022asymptotic, and
kirch2022sequential). Thus, as a second contribution, we build a
composite online detection scheme. Our method is inspired by the popular
\textquotedblleft majority vote\textquotedblright\ classifiers (see the
review by dietterich2000ensemble), where different classification
rules are combined into one rule based e.g. on the mode (or
\textquotedblleft majority vote\textquotedblright ) thereof. We propose a
\textquotedblleft veto-based\textquotedblright\ changepoint learning
procedure, where, in essence, at each point in time $t\geq m+1$ the CUSUM\
process is simultaneously weighted using several weighing schemes, with a
changepoint being flagged up as long as the null is rejected for at least
one set of weights. Heuristically, this procedure should yield the fastest
changepoint detection, irrespective of the location of the changepoint,
whilst still offering size control. Referring to the papers by
kirch2018modified, yu2020note, and romano2023fast
mentioned above, in essence the sequential detection methodology they
investigate is based on using a combination of CUSUM functionals with
different windows, which is highly effective but comes at a relatively high
computational cost. In our case, we similarly propose the combination of
several statistics constructed with different specifications, but at a much
lower computational cost.
The remainder of the paper is organised as follows. We present our model and
the main assumptions, in Section (ref). R\'{e}nyi statistics are
studied in Section (ref); veto-based statistics are in Section (ref). We report a comprehensive Monte Carlo exercise in Section (ref).
Section (ref) concludes.
NOTATION. Throughout the paper, positive, finite constants are denoted as $
c_{0}$, $c_{1}$, ... and their value can change from line to line. We denote
vectors using lower case bold face (e.g. $\mathbf{x}$) and their $j$-th
coordinate as $x_{j}$; matrices are denoted using upper case bold face (e.g.
$\mathbf{A}$) and their element in position $\left( j,h\right) $ is denoted
as $\mathbf{A}_{j,h}$. We use $\left\Vert \mathbf{x}\right\Vert $ to
indicate the Euclidean norm of $\mathbf{x}$; $\left\Vert \mathbf{A}
\right\Vert _{F}$ to indicate the Frobenius norm of a matrix $\mathbf{A}$;
and $s_{i}\left( \mathbf{A}\right) $ is the $i$-th largest singular value of
$\mathbf{A}$, with the largest and smallest singular values denoted as $
s_{\max }\left( \mathbf{A}\right) $\ and $s_{\min }\left( \mathbf{A}\right) $
\ respectively. We use: \textquotedblleft $\rightarrow $\textquotedblright\
to denote the ordinary limit; \textquotedblleft $\overset{a.s.}{\rightarrow }
$\textquotedblright\ for almost sure convergence; \textquotedblleft $\overset
{\mathcal{P}}{\rightarrow }$\textquotedblright\ for convergence in
probability; \textquotedblleft $\overset{\mathcal{D}}{\rightarrow }$
\textquotedblright\ for weak convergence; \textquotedblleft $\overset{
\mathcal{D}}{=}$\textquotedblright\ to indicate equality in distribution.
Given a random variable $X$, we denote its $L_{{\Greekmath 0117} }$-norm as $\left\vert
X\right\vert _{{\Greekmath 0117} }=\left( E\left\vert X\right\vert ^{{\Greekmath 0117} }\right) ^{1/{\Greekmath 0117}
} $. We use the symbol $\Omega \left( \cdot \right) $ to indicate the exact
order of magnitude of a sequence, i.e. $s_{m}=\Omega \left( m^{{\Greekmath 0115}
}\right) $ means that $\lim_{m\rightarrow \infty }m^{-{\Greekmath 0115} }s_{m}=c_{0}$
with $0<c_{0}<\infty $. Finally, $\left\{ W\left( u\right) ,u\geq 0\right\} $
\ denotes a standard Wiener process. Other, relevant notation is introduced
later on in the paper.
Model and assumptions
In this section, we introduce our main model and assumptions. Our workhorse
model is the linear regression model of equation ((ref)), viz.
equation*[equation* omitted — 93 chars of source]
where $\mathbf{x}_{t}$ is a $d$-dimensional vector of covariates with
coordinates denoted as $x_{j,t}$, $1\leq j\leq d$, including a constant -
viz. $x_{1,t}=1$, $t\geq 1$. In ((ref)), the regression
coefficients ${\Greekmath 010C} _{t}$ may be time-varying. We begin with considering a
static model, where no lagged values of $y_{t}$ are used; in Section (ref), we extend our results to the dynamic model. As a final remark, (
(ref)) naturally nests the standard location model, where $d=1$,
and sequential monitoring boils down to monitoring for shifts in the mean of
$y_{t}$.
We now list the main assumptions. We begin by providing a definition of $
L_{{\Greekmath 0117} }$-decomposable Bernoulli shifts.
definitionThe sequence $\left\{ m_{t},-\infty <t<\infty \right\} $
forms an $L_{{\Greekmath 0117} }$-decomposable Bernoulli shift if and only if $
m_{t}=h\left( {\Greekmath 0111} _{t},{\Greekmath 0111} _{t-1},...\right) $, where: $\left\{ {\Greekmath 0111}
_{t},-\infty <t<\infty \right\} $ is an i.i.d. sequence with values
in a measurable space $S$; $h\left( \cdot \right) :S^{\mathbb{N}}\rightarrow
\mathbb{R}
$ is a non random measurable function; $\left\vert m_{t}\right\vert _{{\Greekmath 0117}
}<\infty $; and $\left\vert m_{t}-\widetilde{m}_{t,\ell }\right\vert _{{\Greekmath 0117}
}\leq c_{0}\ell ^{-a}$, for some $c_{0}>0$ and $a>0$, where $\widetilde{m}
_{t,\ell }=h\left( {\Greekmath 0111} _{t},...,{\Greekmath 0111} _{t-\ell +1},\widetilde{{\Greekmath 0111} }_{t-\ell
,t,\ell },\right. $ $\left. \widetilde{{\Greekmath 0111} }_{t-\ell -1,t,\ell }...\right) $
, with $\left\{ \widetilde{{\Greekmath 0111} }_{s,t,\ell },-\infty <s,\ell ,t<\infty
\right\} $ i.i.d. copies of ${\Greekmath 0111} _{0}$, independent of $\left\{
{\Greekmath 0111} _{t},-\infty <t<\infty \right\} $.
Since the seminal works by wu2005nonlinear and
berkes2011split, decomposable Bernoulli shifts have proven a
convenient way to model dependent time series, mainly due to their
generality and to the fact that it is much easier to verify whether a
sequence forms a decomposable Bernoulli shift than e.g. verifying mixing
conditions. Virtually all the most common DGPs in econometrics and
statistics can be shown to satisfy Definition (ref):
liu2009strong, inter alia, provide various theoretical
results, and numerous examples including ARMA models, ARCH/GARCH sequences,
and other nonlinear time series models (such as e.g. random coefficient
autoregressive models and threshold models). Moreover, it is easy to verify
that if $\left\{ m_{t},-\infty <t<\infty \right\} $ is an $L_{{\Greekmath 0117} }$
-decomposable Bernoulli shift, then $\left\{ m_{t}^{{\Greekmath 0114} },-\infty
<t<\infty \right\} $ is an $L_{{\Greekmath 0117} /{\Greekmath 0114} }$-decomposable Bernoulli shift.
Hence, the weak dependence of Definition (ref) is sufficiently
flexible to allow to study changes in the mean, and also in higher order
moments, of $m_{t}$.
We are now ready to present our assumptions.
assumption(i) $\left\{ {\Greekmath 010F} _{t},-\infty <t<\infty \right\} $
forms an $L_{4}$-decomposable Bernoulli shift with $E\left( {\Greekmath 010F}
_{t}\right) =0$, $\left\vert {\Greekmath 010F} _{t}\right\vert _{2}>0$, and $a>2$;
(ii) for $2\leq j\leq d$, $\left\{ x_{j,t},-\infty <t<\infty \right\} $
forms an $L_{4}$-decomposable Bernoulli shift with $a>2$.
assumption(i) $E\left( {\Greekmath 010F} _{t}\right) =0$; (ii) $E\left(
x_{j,t}{\Greekmath 010F} _{t}\right) =0$, for all $2\leq j\leq d$; (iii) $E\left(
\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }\right) =\mathbf{C}$, where $\mathbf{C}
$ is such that $0<s_{\min }\left( \mathbf{C}\right) \leq s_{\max }\left(
\mathbf{C}\right) <\infty $.
Assumption (ref) requires that both the error term ${\Greekmath 010F} _{t}$
and the regressors $\mathbf{x}_{t}$ admit at least four moments. In essence,
this assumption entails the validity of various limit theorems, and in
particular of a uniform estimate of the convergence rate (also known as
\textquotedblleft strong approximation\textquotedblright ) of the invariance
principle (see Lemma (ref)). In principle, it would be possible to
reduce the moment existence requirement, upon assuming independence between $
{\Greekmath 010F} _{t}$ and $\mathbf{x}_{t}$. As far as serial dependence is
concerned, the requirement that $a>2$ is quite mild; indeed the DGPs
typically used in statistics and econometrics have an exponential
rate of decay for $\left\vert m_{t}-\widetilde{m}_{t,\ell }\right\vert _{{\Greekmath 0117}
}$. As far as Assumption (ref) is concerned,\ parts (i)
and (ii) are standard; part (iii) rules out collinearity,
by stating that the smallest eigenvalue of $\mathbf{C}$ is bounded away from
zero.
Sequential detection of changepoints
Our statistics are based on comparing the regression coefficients estimated
at each point in time over the monitoring horizon against a benchmark
estimate obtained during a training period $1\leq t\leq m$ in which no break
was observed.
assumption${\Greekmath 010C} _{t}={\Greekmath 010C} _{0}$, for all $1\leq t\leq m$.
After $m$, the monitoring procedure starts, for the null hypothesis of ((ref)). Let $\widehat{{\Greekmath 010C} }_{m}$ be the LS\ estimator using data in the
training sample, viz.
equation[equation omitted — 183 chars of source]
and define the LS residuals
equation[equation omitted — 214 chars of source]
Heuristically, if no changepoint is present, $\widehat{{\Greekmath 010C} }_{m}$ is a
valid estimator of ${\Greekmath 010C} _{0}$ throughout the monitoring period; hence, it
can be expected that $\widehat{{\Greekmath 010F} }_{t}$ will fluctuate across zero
for all $t\geq m+1$. Conversely, in the presence of a break, this creates a
bias term in $\widehat{{\Greekmath 010F} }_{t}$. Hence, a natural way of testing for
changepoints is to use, as detector, a CUSUM-type statistic defined as the
partial sums process of the residuals over the monitoring horizon, i.e.
equation[equation omitted — 184 chars of source]
We assume that the monitoring does not go on forever, but it is carried out
over a horizon of length, say, $T_{m}$, after which it is terminated.
assumption$\lim_{m\rightarrow \infty }T_{m}=\infty $.
Note that, in Assumption (ref), the monitoring horizon can be
\textquotedblleft long\textquotedblright , i.e. we allow for $T_{m}=\Omega
\left( m^{{\Greekmath 0115} }\right) $ with ${\Greekmath 0115} \geq 1$, but it can also be
\textquotedblleft short\textquotedblright , i.e. $T_{m}=o\left( m\right) $;
the latter case is seldom considered in the literature. Under such a
closed-ended monitoring scheme, our null hypothesis of interest in ((ref)) becomes
equation[equation omitted — 186 chars of source]
In practice, $Q\left( m;k\right) $ is compared against a function of $k$,
and, as soon as it exceeds it, a changepoint is detected. Such a function is
called the boundary function; its specification is important in
order to ensure the procedure-wise control of Type I Errors under $H_{0}$
and, on the other hand, to ensure power and a timely detection in the
presence of a break. horvath2004monitoring and
horvath2007sequential suggest using, in ((ref)), a family
of boundary function indexed by a user-chosen parameter $0\leq {\Greekmath 0111} \leq 1/2$
, defined as
equation[equation omitted — 166 chars of source]
where
equation[equation omitted — 146 chars of source]
is the long-run variance of $\left\{ {\Greekmath 010F} _{t},-\infty <t<\infty
\right\} $. Hence, a changepoint is marked according to the decision rule
equation[equation omitted — 495 chars of source]
where $c_{{\Greekmath 010B} ,{\Greekmath 0111} }$ is a critical value for a pre-specified,
user-defined nominal size ${\Greekmath 010B} $. In ((ref)), intuitively, the
term $m^{1/2}\left( 1+k/m\right) $ is a norming sequence which ensures that $
Q\left( m;k\right) $ is properly normed; conversely, $\left( k/\left(
m+k\right) \right) ^{{\Greekmath 0111} }$ is a weight function. Heuristically, the larger
${\Greekmath 0111} $, the smaller the boundary function $g_{{\Greekmath 0111} }(m,k)$, and therefore
the higher the detection ability of our monitoring scheme.
aue2004delay and aue2009delay prove that this intuition is
correct, by showing that the detection delay in the presence of a break is
inversely related to ${\Greekmath 0111} $. Building on this intuition, here we study ((ref)) with ${\Greekmath 0111} \in (1/2,\infty )$.
Asymptotics under the null for R\'{e}nyi statistics
In order to study the limiting distribution under the null, note that ((ref)) states that the event $\left\{ {\Greekmath 011C} _{m}<T_{m}\right\} $ is
tantamount to having
equation*[equation* omitted — 157 chars of source]
Using the Law of the Iterated Logarithm, it can be shown that $\max_{1\leq
k\leq T_{m}}\left\vert Q\left( m;k\right) \right\vert \geq c_{{\Greekmath 010B} ,{\Greekmath 0111}
}g_{{\Greekmath 0111} }(m;k)\overset{a.s.}{\rightarrow }\infty $; however, we can derive
a well-defined limit for
equation*[equation* omitted — 170 chars of source]
for a user-specified trimming sequence $a_{m}$ such that
equation[equation omitted — 223 chars of source]
In ((ref)), $a_{m}$ needs to diverge as $m\rightarrow \infty $, but this
can happen at an arbitrarily slow rate.\ From an operational viewpoint, this
entails that our monitoring scheme cannot start straight after $m$, but only
after $m+a_{m}$ periods, so that ((ref)) modifies into
equation[equation omitted — 479 chars of source]
of course, $a_{m}$ cannot be longer than $T_{m}$.
Define the norming sequence $r_{m}=a_{m}/\left( a_{m}+m\right) $. The
following theorem provides the limiting distribution of our statistics under
the null hypothesis.
theoremUnder Assumptions (ref)-(ref) and ((ref)), as $m\rightarrow \infty $ with $d=O\left( m^{1/4}\right) $, it holds
that, for all ${\Greekmath 0111} >1/2$
\begin{equation*}
r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
Q(m;k)\right\vert }{g_{{\Greekmath 0111} }(m;k)}\overset{\mathcal{D}}{\rightarrow }
\sup_{1\leq u<\infty }\frac{\left\vert W(u)\right\vert }{u^{{\Greekmath 0111} }}.
\end{equation*}
Theorem (ref) provides the weak limit of the ratio between the
detector $Q(m;k)$ and the boundary function $g_{{\Greekmath 0111} }\left( m;k\right) $,
thus offering a way of computing asymptotic critical values. We are not
aware of any closed form expression for the quantiles of $\sup_{1\leq
u<\infty }u^{-{\Greekmath 0111} }|W(u)|$, which therefore must be computed by simulation.
However, when ${\Greekmath 0111} \leq 1$, exploiting the scale transformation of the
Wiener process we obtain
equation*[equation* omitted — 267 chars of source]
in such a case, the quantiles of $\sup_{1\leq u<\infty }u^{-{\Greekmath 0111} }|W(u)|$
are the same as those in Table 1 in horvath2004monitoring, computed
for the case $1-{\Greekmath 0111} $.
As far as $a_{m}$ is concerned, it would be desirable to have $a_{m}$ as
small as possible, so as to ensure a fast detection of breaks occurring
close to the beginning of the monitoring horizon. Also, Theorem (ref) requires the restriction $d=O\left( m^{1/4}\right) $. This can
be read in two ways: on the one hand, given $m$, it poses a limit on the
dimension of $\mathbf{x}_{t}$ the size of ((ref)), indicating
that some dimension reduction must be carried out prior to monitoring ((ref)). On the other hand, the size of the training sample $m$ can be
viewed as a tuning parameter whose choice depends, inter alia, on $d$.
We conclude by noting that, in order to make our statistics feasible, an
estimator of the long-run variance ${\Greekmath 011B} ^{2}$ is needed. Based on
Assumption (ref), we propose a standard,
weighted-sum-of-covariances estimator, computed using the full training
sample. Seeing as no breaks occur between $1\leq t\leq m$, such an estimator
can be expected to be consistent under both the null and the alternative.
Let $\widehat{{\Greekmath 010D} }_{j}=m^{-1}\sum_{t=j+1}^{m}\widehat{{\Greekmath 010F} }_{t}
\widehat{{\Greekmath 010F} }_{t-j}$; we propose the following Bartlett-type estimator
equation[equation omitted — 182 chars of source]
where $H$ is a bandwidth parameter. Other estimators are possible, and we
refer to andrews1991heteroskedasticity and casini2023theory\
for useful references.
lemmaUnder Assumptions (ref)-(ref), it holds
that
\begin{equation*}
\widehat{{\Greekmath 011B} }_{m}^{2}-{\Greekmath 011B} ^{2}=O_{P}\left( \frac{1}{H}\right)
+O_{P}\left( \frac{dH}{m^{1/2}}\right) .
\end{equation*}
Lemma (ref) indicates that, up to an appropriate choice of the bandwidth
$H$, $\widehat{{\Greekmath 011B} }_{m}^{2}$ is a consistent estimator of ${\Greekmath 011B} ^{2}$.
The lemma suggests that a possible choice, designed to balance the
traditional \textquotedblleft bias\textquotedblright\ term of order $
O_{P}\left( 1/H\right) $ and the \textquotedblleft
variance\textquotedblright\ term of order $O_{P}\left( dH/m^{1/2}\right) $,
is $H=O\left( d^{-1/2}m^{1/4}\right) $.
Monitoring schemes based on R\'{e}nyi statistics under
alternatives
Under the alternative, we assume that there is a changepoint at $k^{\ast }$:
equation[equation omitted — 243 chars of source]
We allow the break size to depend on the sample size $m$, thus defining $
\Delta _{m}={\Greekmath 010C} _{A}-{\Greekmath 010C} _{0}$.
theoremWe assume that Assumptions (ref)-(ref), ((ref)) and $\mathbf{c}_{1}^{\prime }\Delta _{m}\neq 0$, are satisfied.
Assume further that either: (i) $k^{\ast }=o\left( a_{m}\right) $ and
\begin{equation}
\liminf_{m\rightarrow \infty }a_{m}^{1/2}\mathbf{c}_{1}^{\prime }\Delta
_{m}=\infty ;
\end{equation}
or (ii) $\liminf_{m\rightarrow \infty }a_{m}^{-1}k^{\ast }>0$, $
\liminf_{m\rightarrow \infty }T_{m}/k^{\ast }>1$\ and
\begin{equation}
\liminf_{m\rightarrow \infty }r_{m}^{{\Greekmath 0111} -1/2}m^{1/2}\left( \frac{k^{\ast }
}{m+k^{\ast }}\right) ^{1-{\Greekmath 0111} }\mathbf{c}_{1}^{\prime }\Delta _{m}=\infty .
\end{equation}
Then it holds that, for all ${\Greekmath 0111} >1/2$, $\lim_{m\rightarrow \infty }P\left(
{\Greekmath 011C} _{m}<T_{m}|H_{A}\right) =1$.
Part (i) of Theorem (ref) states that our statistics have
power versus a changepoint occurring close to the beginning of the
monitoring horizon. Interestingly, under ((ref)), breaks can be
\textquotedblleft small\textquotedblright , and the case $\left\Vert \Delta
_{m}\right\Vert \rightarrow 0$ as $m\rightarrow \infty $\ can be considered;
however, breaks cannot be \textquotedblleft too small\textquotedblright .
Conceptually similar results are derived in horvath2022changepoint
for the case of offline changepoint detection. According to part (ii)
of the theorem, tests also have power versus later occurring breaks, i.e.
under $\liminf_{m\rightarrow \infty }a_{m}^{-1}k^{\ast }>0$ and ((ref)). For example, considering the case of $k^{\ast }=O\left(
m\right) $, condition ((ref)) boils down to
equation[equation omitted — 213 chars of source]
which indicates that, whenever ${\Greekmath 0111} \leq 1$, detection will take place.
However, upon carefully studying the proof of Theorem 2.2 in
horvath2004monitoring, it follows that whenever ${\Greekmath 0111} <1/2$
changepoints are detected as long as $m^{1/2}\left\vert \mathbf{c}
_{1}^{\prime }\Delta _{m}\right\vert \rightarrow \infty $, which is more
easily satisfied than ((ref)), and allows for smaller
changepoint magnitudes. Hence, whilst ensuring power versus early
changepoints, the use of heavy weights in the boundary function $g_{{\Greekmath 0111}
}\left( m;k\right) $ reduces the ability to detect changepoints occurring
later on during the monitoring horizon. This can be seen by considering the
case of ${\Greekmath 0111} >1$ in ((ref)), or the case where $
m^{-1}k^{\ast }\rightarrow \infty $. In the latter case, condition
(ii) is tantamount to
equation[equation omitted — 192 chars of source]
where again values of ${\Greekmath 0111} >1$ are detrimental to the power of the
monitoring scheme. Finally, although we have focussed on the case of an
abrupt break, following our proofs it is possible to verify that our
monitoring schemes have power also in the presence of smooth breaks.
Theorem (ref) indicates that our statistics seem particularly suited
to the detection of early changepoints. We now study this case more
in-depth, considering the asymptotics for the detection delay in the
presence of early occurring changes.
assumption(i) $k^{\ast }=o\left( a_{m}\right) $; (ii) $
\left\Vert \mathbf{c}_{1}\right\Vert =\Omega \left( d^{1/2}\right) $; (iii)
as $m\rightarrow \infty $, $a_{m}^{1/2}d^{1/2}\left\Vert \Delta
_{m}\right\Vert \rightarrow \infty $.
Assumption (ref) deals with the location and the size of
the changepoint. According to part (i), the break is assumed to
occur early, before the start of the monitoring procedure. This can be read
in conjunction with similar results derived in aue2004delay and
aue2009delay, where the asymptotic distribution of ${\Greekmath 011C} _{m}$ is
derived under the assumption of an early break.
theoremUnder Assumptions (ref)-(ref) and (
(ref)), it holds that ${\Greekmath 011C} _{m}\overset{\mathcal{P}}{\rightarrow }a_{m}$
.
Theorem (ref) states that, if a break occurs prior to the trimming
point $a_{m}$, then it is detected immediately at the start of the
monitoring procedure. This result offers an important insight on the
relevance and practical use of our statistics. The theory derived in
aue2004delay and aue2008monitoring indicates that using $
{\Greekmath 0111} \leq 1/2$ ensures detection of changepoints occurring not too soon
after $m$; in particular, using ${\Greekmath 0111} =1/2$ ensures a short delay - of order
$O\left( \sqrt{\ln \ln m}\right) $ - in detecting breaks occurring $o\left(
\sqrt{\ln \ln m}\right) $ periods from $m$. Based on this and on Theorem (ref), in the case of our statistics, the delay is of order $a_{m}$ for
breaks occurring $o\left( a_{m}\right) $ periods from $m$: upon choosing $
a_{m}=o\left( \sqrt{\ln \ln m}\right) $, this entails that early occurring
breaks are found even more quickly than when using ${\Greekmath 0111} =1/2$, with the
added advantage that one does not need to use the asymptotic critical values
computed for the case ${\Greekmath 0111} =1/2$, where convergence is notoriously slow.
Extensions to the case of dynamic regressors
For the ease of exposition, in the above we have considered the case of
exogenous regressors only. We now extend our results to the case of a
dynamic model:
equation[equation omitted — 235 chars of source]
where $\mathbf{y}_{p,t}^{\prime }=\left( y_{t-1},...,y_{t-p}\right) ^{\prime
}$, ${\Greekmath 010C} _{t}^{Y}=\left( {\Greekmath 011A} _{1},...,{\Greekmath 011A} _{p}\right) ^{\prime }$, $
y_{0} $ is an initial condition, and $\mathbf{z}_{t}$ is a $\left(
d-p\right) $-dimensional vector of exogenous regressors, and $\mathbf{x}
_{t}=\left( \mathbf{z}_{t}^{\prime },\mathbf{y}_{p,t}^{\prime }\right)
^{\prime }$ is - by construction - of dimension $d$. With this notation, we
are able to define the null hypothesis of interest ((ref)), the
alternative hypothesis ((ref)), and the monitoring
statistics, exactly in the same way as above.
assumption(i) $\left\vert y_{0}\right\vert _{4}<\infty $; (ii) the
polynomial equation $1-\sum_{j=1}^{p}{\Greekmath 011A} _{j}a^{j}=0$, with $a\in \mathbb{C}
$, has all roots outside the unit circle.
theoremUnder e assume that the data are generated by ((ref)) and Assumption (ref), the results in Theorems (ref), (ref) and (ref), and Lemma (ref), hold under
the same assumptions.
Veto-based changepoint learning
Our theory, and the theory developed in other contributions (e.g.
aue2004delay and aue2008monitoring), suggests that
different values of ${\Greekmath 0111} $ work better for different changepoint locations.
Hence, any sequential monitoring methodology based on a specific ${\Greekmath 0111} $ can
be expected to offer superior detection timeliness in some cases, and to be
dominated by different choices of ${\Greekmath 0111} $ in other cases; this issue is also
echoed in a recent contribution by kirch2022sequential.
In this section, we consider a combination of several detection rules. Upon
inspecting our proofs and the proofs in horvath2004monitoring, it
can be shown that test statistics using ${\Greekmath 0111} <1/2$ and ${\Greekmath 0111} >1/2$ are
asymptotically independent of each other,\footnote{
Intuitively, this is due to the facts that: (1) the limiting distribution
when ${\Greekmath 0111} <1/2$ is determined by the \textquotedblleft
central\textquotedblright\ part of $\left\{ W\left( u\right) ,u\geq
0\right\} $, whereas, when using ${\Greekmath 0111} >1/2$, the limit is completely
determined by the \textquotedblleft early\textquotedblright\ part of $
\left\{ W\left( u\right) ,u\geq 0\right\} $ (see also the comments at the
end of the proof of Theorem (ref)); and (2) that the Wiener
process has independent increments.} although of course two or more
statistics with different ${\Greekmath 0111} >1/2$ are not independent of each other (the
same holds for statistics with different ${\Greekmath 0111} <1/2$). Hence, it is in
principle possible to combine different test statistics, although using a
combination of tests, e.g. with the Bonferroni correction, is bound to be
subject to the typical criticisms of being overconservative and, therefore,
liable to not detect (timely, or at all) a changepoint. Hence, in the light
of the considerations above, and in order to ensure the fast changepoint
detection whilst controlling for false positives, we propose a different,
composite sequential detection scheme. We base our methodology on a
\textquotedblleft veto-based\textquotedblright\ approach, where several
statistics (corresponding to different choices of ${\Greekmath 0111} $) are employed
simultaneously, and the null is rejected as long as at least one statistic
exceeds its critical value. Heuristically, the veto-based decision rule
offers a compromise between the various choices of ${\Greekmath 0111} $, ensuring that
detection delay is good irrespective of the changepoint location.
Recall the detector $Q\left( m;k\right) $ defined in ((ref)). For each
choice of ${\Greekmath 0111} _{j}$, $j=1,2,...,J$, we have a corresponding boundary
function $g_{{\Greekmath 0111} _{j}}(m;k)$, and a critical value at nominal level ${\Greekmath 010B}
$, $c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}$, with rejection occurring at the first value of $
k\geq 1$ such that $\left\vert Q\left( m;k\right) \right\vert >c_{{\Greekmath 010B}
,{\Greekmath 0111} _{j}}g_{{\Greekmath 0111} _{j}}(m;k)$. We combine several statistics using $0\leq
{\Greekmath 0111} _{j}\leq 1$; recall also that, when ${\Greekmath 0111} _{j}>1/2$, the monitoring
starts after $a_{m}$ periods from the beginning of the $1\leq k\leq T_{m}$
horizon. Hence, the veto-based procedure stops at $\widetilde{{\Greekmath 011C} }_{m}$
defined as
equation[equation omitted — 665 chars of source]
where
equation*[equation* omitted — 295 chars of source]
and
equation*[equation* omitted — 476 chars of source]
In ((ref)), $C_{{\Greekmath 010B} }$ is the procedure-wise critical value,
such that $\lim_{m\rightarrow \infty }P\left( \widetilde{{\Greekmath 011C} }
_{m}<T_{m}|H_{0}\right) ={\Greekmath 010B} $.
In order to implement ((ref)), we require an asymptotic
approximation of the procedure-wise critical value $C_{{\Greekmath 010B} }$. Define $
\widetilde{{\Greekmath 0111} }_{j}={\Greekmath 0111} _{j}I\left( {\Greekmath 0111} _{j}\leq 1/2\right) +\left(
1-{\Greekmath 0111} _{j}\right) I\left( {\Greekmath 0111} _{j}>1/2\right) $, and modify Assumption (ref) to consider a long monitoring horizon.
assumption(i) $\lim_{m\rightarrow \infty }T_{m}=\infty $; (ii) $
\liminf_{m\rightarrow \infty }T_{m}/m>0$; (iii) $T_{m}=\Omega \left(
m^{{\Greekmath 0115} }\right) $, with ${\Greekmath 0115} \geq 1$.
Assumption (ref) is required to derive the asymptotics of
monitoring statistics with ${\Greekmath 0111} <1/2$ (see horvath2004monitoring).
The following results characterise the null distribution, and the power
under alternatives, of veto-based statistics.
theoremWe assume that Assumptions (ref)-(ref) and (ref), ((ref)) and $d=o\left(
m^{1/4}\right) $, are satisfied. Then, under the null, as $m\rightarrow
\infty $\ it holds that
\begin{equation}
\max_{1\leq k\leq T_{m}}\frac{\left\vert Q\left( m;k\right) \right\vert }{
\min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }g_{{\Greekmath 0111} _{j}}(m;k)}\overset{\mathcal{D}}{\rightarrow }
\sup_{0<u<1}\frac{\left\vert W\left( u\right) \right\vert }{\min_{1\leq
j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{\widetilde{{\Greekmath 0111} }_{j}}}.
\end{equation}
theoremWe assume that the assumptions of Theorem (ref)
hold, and further that either: (i) $k^{\ast }=o\left( a_{m}\right) $ and (
(ref)); or (ii) $\liminf_{m\rightarrow \infty }a_{m}^{-1}k^{\ast
}>0 $ and ((ref)) hold; or (iii) that $k^{\ast }=O\left( m\right) $
and
\begin{equation}
\liminf_{m\rightarrow \infty }\max_{1\leq j\leq J}\frac{\left( k^{\ast
}\right) ^{1-{\Greekmath 0111} _{j}}}{\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}m^{1/2-{\Greekmath 0111} _{j}}}
\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert =\infty .
\end{equation}
Then it holds that
\begin{equation}
\lim_{m\rightarrow \infty }P\left( \widetilde{{\Greekmath 011C} }_{m}<T_{m}|H_{A}\right)
=1.
\end{equation}
Some comments are in order. Theorem (ref) stipulates that,
asymptotically
equation[equation omitted — 262 chars of source]
hence, $C_{{\Greekmath 010B} }$ can be obtained by standard Monte Carlo techniques.
According to Theorem (ref), the veto-based decision rule does
have power versus changepoints; note that the proposition studies only the
case $k^{\ast }=O\left( m\right) $ - i.e. breaks occurring not too late
during the monitoring period - but it could be extended to the case where $
k^{\ast }$ is \textquotedblleft bigger\textquotedblright\ than $m$ after
elementary, if tedious, algebra. Condition ((ref)) is a
high-level requirement which can be more easily interpreted by considering
some leading examples. To begin with, we investigate the case $k^{\ast
}=c_{0}m$, corresponding to a break occurring later on during the monitoring
horizon. In such a case, a possible question is how small can the size of
the break, $\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert $, be.
When ${\Greekmath 0111} _{j}<1/2$, the non-centrality parameter in ((ref))
is proportional to $m^{1/2}\left\vert \mathbf{c}_{1}^{\prime }\Delta
_{m}\right\vert $, which entails that nontrivial power is found versus
breaks of magnitude at least $O\left( m^{-1/2}\right) $; note that this
holds irrespective of the actual value of ${\Greekmath 0111} _{j}$. On the other hand,
when ${\Greekmath 0111} _{j}>1/2$, ((ref)) is proportional to $a_{m}^{{\Greekmath 0111}
_{j}-1/2}m^{1-{\Greekmath 0111} _{j}}\left\vert \mathbf{c}_{1}^{\prime }\Delta
_{m}\right\vert $, suggesting that the larger the choice of ${\Greekmath 0111} _{j}$, the
larger $\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert $ needs to
be to be detected. In this case, the veto-based rule will be triggered by
the statistics with ${\Greekmath 0111} _{j}<1/2$, thus ensuring power versus small breaks.
In conclusion, we would like to point out that an alternative approach to
the \textquotedblleft classical\textquotedblright\ CUSUM detectors employed
here, is the so-called Page-CUSUM detectors (see e.g.
fremdt2015page, kirch2018modified, and
romano2023fast), defined as
equation[equation omitted — 209 chars of source]
HT2023 consider the weighted version of ((ref)). Intuitively,
at each point in time $1\leq k\leq T_{m}$, the detector is set equal to the
\textquotedblleft worst case\textquotedblright\ observed up to $k$. This
should make the detector $Q^{\dagger }(m;k)$ more sensitive, and more prone
to using only the observations which, up to time $k$, are affected by a
possible changepoint. Indeed, the simulations in HT2023 show that
the use of the Page-CUSUM detector does offer some improvement on the
standard CUSUM, but the main gains can be ascribed to the use of heavier and
heavier weighing schemes. Furthermore, the use of Page-CUSUM methodologies
is marred by their computational complexity (see however the solution
proposed in a recent contribution by romano2023fast). In this
respect, the veto-based approach suggested in this section is similar, in
spirit, to ((ref)); instead of using the \textquotedblleft worst case
scenario\textquotedblright\ across $k$, we use the (computationally simpler)
\textquotedblleft worst case scenario\textquotedblright\ across several
values of ${\Greekmath 0111} $.
comment\subsubsection{Veto-based procedure with size sacrifice}
Simulations show that the asymptotic critical values computed according to
Proposition (ref), $C_{{\Greekmath 010B} }$, are (slightly) larger than $1$
across various values of ${\Greekmath 010B} $. This entails that the veto-based
procedure does not pick the best (delay-minimizing) weight ${\Greekmath 0111} _{j}$
depending on the location $k^{\ast }$ of the changepoint, but rather it
finds a compromise among weights in order to avoid massive delays. In this
respect, our veto-based procedure is more akin to a \textquotedblleft
negotiation-based procedure\textquotedblright .
In order to further improve on this, we make the following heuristic
considerations. The choice of $C_{{\Greekmath 010B} }$ based on ((ref))
ensures asymptotic size control; however, the notion of size in the context
of sequential monitoring is different to the notion of size control in a
typical Neyman-Pearson paradigm. In the latter context, a test is required
to have a pre-specified nominal size equal to ${\Greekmath 010B} $; hence, the
goal of using asymptotic critical values should ensure that, under the null,
empirical rejection frequencies are around ${\Greekmath 010B} $. Conversely,
in the context of sequential monitoring, the goal is to ensure that Type I
Error do not occur often at a procedure-wise level; thus, asymptotic
critical values are designed in order to ensure that, under the null,
empirical rejection frequencies are below ${\Greekmath 010B} $. In turn, this
entails that sequential monitoring is bound to be conservative and
undersized; hence, under alternatives, sequential monitoring is likely to
have lower power (and higher delays) than optimal. On account of these
heuristic consideration, a possible approach to enhance power and reduce
delays could be to \textquotedblleft shave off\textquotedblright\ $C_{{\Greekmath 010B}
}$, thus leveraging on the size/power trade-off and\ gaining some power at
the expense of size. We call this approach a \textquotedblleft veto-based
procedure with size sacrifice\textquotedblright .
Again heuristically, as mentioned above $C_{{\Greekmath 010B} }$ is slightly above $1$.
We propose to set $C_{{\Greekmath 010B} }=1$, which corresponds to a full-blown
veto-based approach. Our simulations suggest that the following rule
represents, across all cases considered, a good compromise between
minimising delay and avoiding oversizement
\begin{equation}
\widetilde{{\Greekmath 011C} }_{m}^{\ast }=
\begin{cases}
\inf \{k\geq 1:|Q\left( m;k\right) |\geq \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }g_{{\Greekmath 0111} _{j}}\left( m;k\right) \}, & \\
T_{m},\ \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}\ |Q\left( m;k\right) |\leq \min_{1\leq j\leq J}c_{{\Greekmath 010B}
,{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }g_{{\Greekmath 0111} _{j}}\left( m;k\right) \ \relax\ifmmode\expandafter\text@\else\expandafter\mbox\fi{for\ all}\
1\leq k\leq T_{m}. &
\end{cases}
\end{equation}
Simulations
We report some Monte Carlo evidence to evaluate the finite sample
performance of our methodology, and to offer guidelines on the choice of
some specifications, chiefly $a_{m}$.\footnote{
Further Monte Carlo evidence is reported in Section (ref) in the
Supplement.} All our experiments are based on the following design
equation[equation omitted — 130 chars of source]
where we allow for the possible presence of a dynamic term. Under the null,
we generate ${\Greekmath 010F} _{t}$ as i.i.d. $N\left( 0,1\right) $ for $
1\leq t\leq m+T_{m}$, and each coordinate of ${\Greekmath 010C} _{0}$ as ${\Greekmath 010C}
_{0,j}\sim 1+{\Greekmath 011B} _{{\Greekmath 010C} }N\left( 0,1\right) $, independently across $
1\leq j\leq d$. Our results are derived under ${\Greekmath 011A} =0.5$ and ${\Greekmath 011B}
_{{\Greekmath 010C} }=0.5$, but unreported experiments showed that the value of ${\Greekmath 011B}
_{{\Greekmath 010C} }$ does not alter our results. We allow for serial dependence in the
$d$-dimensional regressor $\mathbf{x}_{t}$, generating its (demeaned)
coordinates as
equation*[equation* omitted — 61 chars of source]
with ${\Greekmath 011E} =0.5$, and $e_{j,t}\sim N\left( 0,1\right) $, i.i.d.
across $2\leq j\leq d$ and $1\leq t\leq m+T_{m}$ (recall that $x_{1,t}=1$).
In the case of a static regression - i.e., when ${\Greekmath 011A} =0$ in ((ref))\ -
we allow for the regression error ${\Greekmath 010F} _{t}$ to be serially dependent:
equation*[equation* omitted — 89 chars of source]
with $w_{t}\sim N\left( 0,1\right) $ i.i.d. across $1\leq t\leq
m+T_{m}$. We have used ${\Greekmath 0112} =0.5$; again, unreported simulations show
that altering this specification does not have a significant impact on our
result. We estimate the long-run variance using ((ref)) with
bandwidth $H=\left\lfloor m^{2/5}\right\rfloor $. Finally, we set $T_{m}=m$
across all experiments; in the Supplement, we further assess the sensitivity
of our result to the length of the monitoring horizon $T_{m}$.
As far as implementation is concerned, we consider several choices for the
trimming sequence $a_{m}$, studying how these affect the empirical rejection
frequencies and the delays. We also consider several values of ${\Greekmath 0111} $, in
order to assess the impact of this specification on the performance of our
methodologies. By way of comparison, we also use ${\Greekmath 0111} \leq 1/2$; in this
case, we have used the critical values in Table 1 in
horvath2004monitoring. All the results reported here are obtained
for $d=2$ (that is, one exogenous regressor and the constant, in addition to
the lagged dependent variable) and a nominal level ${\Greekmath 010B} =0.05$. Finally,
all simulations were carried out with $2,500$ replications.
table*[table* omitted — 1,883 chars of source]
In Table (ref), we report the empirical rejection frequencies
under the null of no break. In the context of sequential monitoring the
notion of \textquotedblleft size control\textquotedblright\ is different,
and the nominal level ${\Greekmath 010B} $ represents an upper bound for the
probability of a procedure-wise Type I Error.\footnote{
As horvath2007sequential put it, \textquotedblleft \lbrack t]he goal
is to keep the probability of false rejection below ${\Greekmath 010B} $ rather than to
make it close to ${\Greekmath 010B} $\textquotedblright .} Our empirical rejection
frequencies broadly stay beneath the $5\%$ upper bound (with some exceptions
for \textquotedblleft large\textquotedblright\ values of $a_{m}$ and small $
m $), and decline as $m$ increases, indicating that procedure-wise size
control is ensured. The choice of $a_{m}$ is important in determining the
empirical rejection frequencies; as our theory predicts, as $m$ increases
all empirical rejection frequencies fall below the $5\%$ level, but large
values of $a_{m}$ - e.g. corresponding to the choice $a_{m}=\ln ^{2}m$ -
require comparatively higher sample sizes to ensure proper size control. In
Section (ref) in the Supplement, we report more Monte Carlo
evidence, considering also several values of ${\Greekmath 0111} \leq 1/2$ by way of
comparison, on: empirical rejection frequencies under the null with a larger
number of regressors (Tables (ref)-(ref)), showing
that size control, while still guaranteed in general, tends to worsen as $d$
increases; and empirical rejection frequencies under the null using a static
regression model, with various $d$ (Tables (ref)-(ref)), showing that, in this case, the sequential detection
procedure becomes more conservative ceteris paribus compared to a
dynamic model.
table*[table* omitted — 2,147 chars of source]
table*[table* omitted — 2,247 chars of source]
In Tables (ref)-(ref), we report the detection delay
in the presence of a changepoint, considering two types of alternatives: an
early break, occurring exactly $a_{m}$ periods after the start of the
monitoring horizon, and a late break, occurring at $k^{\ast }=m+a_{m}$. We
compare detection delays across several values of ${\Greekmath 0111} $, and for different
choices of $a_{m}$. In the case of early occurring changepoints, sequential
monitoring with ${\Greekmath 0111} >1/2$ ensures a superior performance compared to ${\Greekmath 0111}
\leq 1/2$, as predicted by Theorem (ref). This confirms that using
heavy weights ensures a very fast detection of early occurring breaks. This
is essentially true for all values of ${\Greekmath 0111} $, with the choice ${\Greekmath 0111} =1$
yielding the best results. Interestingly, the choice $a_{m}=\ln \ln m$
delivers the fastest detection, as predicted by the theory, and despite the
fact that in this case the test is undersized. However, on the other hand,
the results in Table (ref) cannot be directly compared across $
a_{m}$, since the location of changepoints is different; for example, when $
a_{m}=\ln ^{2}m$ and therefore also $k^{\ast }=\ln ^{2}m$, this corresponds
to a break occurring early, but not \textquotedblleft very
early\textquotedblright . In this case, detection delays are actually quite
good: Theorem (ref) predicts that breaks are detected after $a_{m}=\ln
^{2}m$ steps, but the results in Table (ref) indicate that such
detection occurs much more quickly (for example, when $m=1000$, it holds
that $\ln ^{2}m\approx 48$, but detection takes place after only $12$
periods for large values of ${\Greekmath 0111} $). This result should further be read in
conjunction with the fact that, according to Table (ref), the
choice $a_{m}=\ln ^{2}m$ results in oversizement for small samples. As far
as early break detection is concerned, results are reversed in the presence
of a late occurring break, as Table (ref)\ shows. In such a
case, delays improve as ${\Greekmath 0111} $ moves towards $1/2$ - both from the left and
the right - but heavily weighted statistics perform substantially worse than
when using ${\Greekmath 0111} \leq 1/2$; this is especially true when considering the
delays under very early occurring breaks in Tables (ref), under $
a_{m}=\ln \ln m$ and even $a_{m}=\ln m$.
In Section (ref) in the Supplement, we report a full-blown of
statistics on detection delays under both alternatives of an early occurring
break and a late one, for various values of $d$, and for both a dynamic
(Tables (ref)-(ref)) and a static (Tables (ref)-(ref)) regression, essentially confirming the
results discussed above.
The distilled essence of Tables (ref)-(ref) is that
- as far as detection delay is concerned - the \textquotedblleft
optimal\textquotedblright\ weight for the CUSUM process, which ensures the
fastest detection irrespective of the changepoint location, does not exist.
Hence, using an agnostic approach such as the one proposed in Section (ref), based on combining various statistics, may be preferable. In the
next set of experiments, we consider several combinations of weighted CUSUM
statistics; in particular, we use a scheme with one lightly weighted and one
heavily weighted statistics (denoted as $\mathcal{V}_{2}$, and based on
using ${\Greekmath 0111} =0.2$ and ${\Greekmath 0111} =0.85$); two lightly weighted and one heavily
weighted statistics (denoted as $\mathcal{V}_{3}$, and based on using ${\Greekmath 0111}
=0.2$, ${\Greekmath 0111} =0.3$ and ${\Greekmath 0111} =0.85$); and two lightly weighted and three
heavily weighted statistics (denoted as $\mathcal{V}_{5}$, and based on
using ${\Greekmath 0111} =0.2$, ${\Greekmath 0111} =0.45$, ${\Greekmath 0111} =0.65$, ${\Greekmath 0111} =0.85$, and ${\Greekmath 0111} =0.9$
).\footnote{
Results with other combinations of ${\Greekmath 0111} $ confirm the findings reported
here, and are available upon request.} We use the same DGP and
specifications as above; by way of comparison, in Table (ref)
we also report empirical rejection frequencies under the null, and
descriptive statistics for the detection delay under the alternative, also
for the cases of individual weighted CUSUM statistics based on ${\Greekmath 0111} =0.25$
and ${\Greekmath 0111} =0.75$.
table*[table* omitted — 1,870 chars of source]
table*[table* omitted — 2,979 chars of source]
Table (ref) states that veto-based detection schemes are able
to afford procedure-wise Type I Error control, especially when $a_{m}$ is
\textquotedblleft small\textquotedblright\ - corresponding to $a_{m}=\ln \ln
m$; indeed, results are very similar to the ones obtained using the
individual, heavily weighted CUSUM statistics. As far as power and timely
detection under alternatives are concerned, Table (ref)
indicates that veto-based statistic guarantee fast detection, both in the
presence of an early break and of a late break. In the former case, the
veto-based procedure delivers essentially the same performance as using a
single, heavily weighted CUSUM statistic, improving on lightly weighted
CUSUM statistics which are less effective; conversely, in the latter case,
the veto-based rules achieve a similar result as lightly weighted CUSUM
statistics, whereas the heavily weighted ones perform poorly in comparison\
- thus confirming the intuition that veto-based detection rules should
combine \textquotedblleft the best of both worlds\textquotedblright . These
results hold, broadly, for all combinations considered, although $\mathcal{V}
_{3}$ and $\mathcal{V}_{5}$ fare better than $\mathcal{V}_{2}$, suggesting
that combining more than two CUSUM\ statistics would result in better
detection in all cases considered. As noted in Tables (ref)-(ref), when using heavily weighted statistics, the best results as
far as detection delays are concerned are found when using $a_{m}=\ln \ln m$
. This finding is important, because the choice of $a_{m}=\ln \ln m$ also
corresponds to the best size control under the null. In the Supplement, we
report more Monte Carlo evidence focusing on the case of a dynamic
regression, studying: empirical rejection frequencies under the null with
various values of $d$\ (Tables (ref) and (ref)),
showing that, as above, size control worsens as $d$ increases, although
using $a_{m}=\ln \ln m$ still offers the best results in all cases
considered; and detection delays under both an early and a late occurring
break with various values of $d$ (Tables (ref) and (ref)), showing that these improve as $d$ increase, mainly as a
consequence of the size/power trade-off.
comment\textquotedblleft Structural changes of uncertain magnitude and
timing have increased the difficulty in forecasting, undermined confidence
in our understanding of the structure of the economy, and increased the risk
of measurement error with respect to key variables.\textquotedblright
Laurence H. Meyer.\footnote{
March 3rd, 2000. Address before the Joint Conference of the Federal Reserve
Bank of San Francisco and the Stanford Institute for Economic Policy
Research, Federal Reserve Bank of San Francisco, San Francisco, California.}
Conclusions and discussion
In this paper, we investigate the well-known, and highly important, issue of
(fast) online changepoint detection. We focus on a linear regression model,
in the possible presence of weak dependence, which also nests the cases of
changepoint detection in the location of a time series $\left\{
y_{t},-\infty <t<\infty \right\} $; moreover, we use a flexible definition
of dependence which allows to readily extend our methods to test for changes
in the higher order moments of $\left\{ y_{t},-\infty <t<\infty \right\} $.
We make two contributions. First, we introduce a class of statistics, called
R\'{e}nyi statistics\ and based on the heavily weighted CUSUM
process of the regression residuals, which are specifically designed to
detect early occurring changepoints. Second, we propose a composite,
veto-based, test statistic which combines different weighted versions of the
CUSUM process of the residuals, designed to ensure timely detection of
changepoints occurring at any point in time over the monitoring horizon. Our
Monte Carlo evidence shows that our statistics offer good control of the
probability of the procedure-wise Type I Error under the null, also
affording timely detection under the alternative. Whilst we focus, as far as
the empirical illustration is concerned, on a macroeconomic example, we
would like to point out that our methodology can be applied to a wide
variety of contexts in virtually all applied sciences.
Several interesting questions remain open and are worth pursuing. In the
construction of heavily weighted CUSUM statistics, the only tuning parameter
is the sequence $a_{m}$ defined in ((ref)). Our simulations suggest that
a good choice to ensure both size control and timely detction is $a_{m}=\ln
\ln m$ in all cases considered; however, having an automated, data-driven
selection rule would be desirable. Heuristically, our proofs (and our
simulations) show that the quality of the asymptotic distribution under the
null is determined by two factors: the speed of convergence in the Gaussian
approximation (see Lemma (ref)), and the impact of $a_{m}$ in the
limiting behaviour of $\max_{a_{m}/m\leq u\leq T_{m}/m}\left\vert W\left(
u/\left( 1+u\right) \right) \right\vert /\left( u/\left( 1+u\right) \right)
^{{\Greekmath 0111} }$ (see the proof of Theorem (ref)). Seeing as $a_{m}$
plays a role only in determining the limit of a heavily weighted Wiener
process, a possible suggestion to the applied user would be to choose the
optimal $a_{m}$ by running a simulation with i.i.d. $N\left(
0,1\right) $ data, generated with training and monitoring samples of size $m$
and $T_{m}$ respectively, determining which choice of $a_{m}$ yields the
best size control without having an overly conservative procedure. As a
second important extension, recently the literature has considered
extensions of online change-point detection methods to different data
structures such as e.g. network data. yu2020note and
dubey2021online study online changepoint detection in networks, and
wang2021optimal extend the CUSUM to the network case. Our
methodologies could lend themselves to being extended to other types of data
structures. These issues have a high priority on the authors' research
agenda.
{ {\
} }
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Further Monte Carlo evidence
Empirical rejection frequencies under the null
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Detection delays
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Veto-based changepoint detection: empirical rejection
frequencies
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Veto-based changepoint detection: detection delays
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Preliminary lemmas
Henceforth, we use the following notation: $\left\lceil {\Greekmath 0125} \right\rceil
$ is the ceiling of a real number ${\Greekmath 0125} $. We also use the convention
that, if a summation involves a non-integer index, this is rounded up - e.g.
$\sum_{i=1}^{{\Greekmath 0125} }=\sum_{i=1}^{\left\lceil {\Greekmath 0125} \right\rceil }$ for
any ${\Greekmath 0125} >1$.
lemmaWe assume that Assumption (ref)(i) holds. Then,
for every $m$, two independent standard Wiener processes $\left\{
W_{1,m}\left( k\right) ,k\geq 1\right\} $ and $\left\{ W_{2,m}\left(
k\right) ,k\geq 1\right\} $ whose distribution does not depend on $m$\ can
be defined on a suitably larger probability space such that, for some $
0<{\Greekmath 0110} _{1}<1/2$, it holds that
\begin{equation}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{1}}}\left\vert
\sum_{t=m+1}^{m+k}{\Greekmath 010F} _{t}-{\Greekmath 011B} W_{1,m}(k)\right\vert =O_{P}(1),
\end{equation}
and
\begin{equation}
\frac{1}{m^{{\Greekmath 0110} _{2}}}\left\vert \sum_{t=1}^{m}{\Greekmath 010F} _{t}-{\Greekmath 011B}
W_{2,m}(m)\right\vert =O_{P}(1),
\end{equation}
where ${\Greekmath 011B} ^{2}$ is defined in ((ref)).
proofThe desired result follows immediately from applying Theorem B.1 in
aue2014dependent; note that the proof of the theorem is based on the
blocking argument, whence the independence of $\left\{ W_{1,m}\left(
k\right) ,k\geq 1\right\} $ and $\left\{ W_{2,m}\left( k\right) ,k\geq
1\right\} $.
lemmaWe assume that Assumptions (ref) and (ref) hold. Then it holds that
\begin{equation*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert
=O_{P}\left( \left( \frac{d}{m}\right) ^{1/2}\right) .
\end{equation*}
proofWe begin by estimating $E\left\Vert \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t}\right\Vert ^{2}$, noting that
\begin{equation}
E\left\Vert \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert
^{2}=\sum_{j=1}^{d}\left\vert \sum_{t=1}^{m}x_{j,t}{\Greekmath 010F} _{t}\right\vert
_{2}^{2}.
\end{equation}
We now show that Assumptions (ref) and (ref) entail that
the sequence $z_{j,t}=x_{j,t}{\Greekmath 010F} _{t}$ is a zero mean, $L_{2}$
-decomposable Bernoulli shift. Indeed, Assumption (ref)
immediately yields $E\left( z_{j,t}\right) =0$; further, consider the
coupling constructions
\begin{align*}
&\widetilde{x}_{j,t,t} =h_{j}^{x}\left( {\Greekmath 0111} _{j,t}^{x},...,{\Greekmath 0111} _{j,1}^{x},
\widetilde{{\Greekmath 0111} }_{j,0,t,t}^{x},\widetilde{{\Greekmath 0111} }_{j,-1,t,t}^{x},...\right) ,
\\
&\widetilde{{\Greekmath 010F} }_{t,t} =h^{{\Greekmath 010F} }\left( {\Greekmath 0111} _{t}^{{\Greekmath 010F}
},...,{\Greekmath 0111} _{1}^{{\Greekmath 010F} },\widetilde{{\Greekmath 0111} }_{0,t,t}^{{\Greekmath 010F} },\widetilde{
{\Greekmath 0111} }_{-1,t,t}^{{\Greekmath 010F} },...\right) ,
\end{align*}
where $\left\{ \widetilde{{\Greekmath 0111} }_{j,0,t,t}^{x},\widetilde{{\Greekmath 0111} }
_{j,-1,t,t}^{x},...\right\} $ are independent copies of $\left\{ {\Greekmath 0111}
_{j,0}^{x},{\Greekmath 0111} _{j,-1}^{x},...\right\} $, and similarly $\left\{ \widetilde{
{\Greekmath 0111} }_{j,0,t,t}^{{\Greekmath 010F} },\widetilde{{\Greekmath 0111} }_{j,-1,t,t}^{{\Greekmath 010F}
},...\right\} $. It holds that
\begin{align*}
x_{j,t}{\Greekmath 010F} _{t} =&\left( x_{j,t}\pm \widetilde{x}_{j,t,t}\right) \left(
{\Greekmath 010F} _{t}\pm \widetilde{{\Greekmath 010F} }_{t,t}\right) \\
=&\widetilde{z}_{j,t,t}+\widetilde{x}_{j,t,t}\left( {\Greekmath 010F} _{t}-\widetilde{
{\Greekmath 010F} }_{t,t}\right) +\widetilde{{\Greekmath 010F} }_{t,t}\left( x_{j,t}-
\widetilde{x}_{j,t,t}\right) +\left( x_{j,t}-\widetilde{x}_{j,t,t}\right)
\left( {\Greekmath 010F} _{t}-\widetilde{{\Greekmath 010F} }_{t,t}\right) ,
\end{align*}
where
\begin{equation*}
\widetilde{z}_{j,t,t}=\widetilde{x}_{j,t,t}\widetilde{{\Greekmath 010F} }_{t,t},
\end{equation*}
is the coupled version of $z_{j,t}$. Hence it follows that
\begin{align*}
\left\vert z_{j,t}-\widetilde{z}_{j,t,t}\right\vert _{2} \leq &\left\vert
\widetilde{x}_{j,t,t}\left( {\Greekmath 010F} _{t}-\widetilde{{\Greekmath 010F} }_{t,t}\right)
\right\vert _{2}+\left\vert \widetilde{{\Greekmath 010F} }_{t,t}\left( x_{j,t}-
\widetilde{x}_{j,t,t}\right) \right\vert _{2}+\left\vert \left( x_{j,t}-
\widetilde{x}_{j,t,t}\right) \left( {\Greekmath 010F} _{t}-\widetilde{{\Greekmath 010F} }
_{t,t}\right) \right\vert _{2} \\
\leq &\left\vert \widetilde{x}_{j,t,t}\right\vert _{4}\left\vert {\Greekmath 010F}
_{t}-\widetilde{{\Greekmath 010F} }_{t,t}\right\vert _{4}+\left\vert x_{j,t}-
\widetilde{x}_{j,t,t}\right\vert _{4}\left\vert \widetilde{{\Greekmath 010F} }
_{t,t}\right\vert _{4}+\left\vert {\Greekmath 010F} _{t}-\widetilde{{\Greekmath 010F} }
_{t,t}\right\vert _{4}\left\vert x_{j,t}-\widetilde{x}_{j,t,t}\right\vert
_{4}
\end{align*}
having used Minkowski's inequality in the first line, and the
Cauchy-Schwartz inequality in the second one. Assumption (ref) now
immediately yields that there exists a constant $0<c_{0}<\infty $ such that
\begin{equation*}
\left\vert z_{j,t}-\widetilde{z}_{j,t,t}\right\vert _{2}\leq c_{0}t^{-a},
\end{equation*}
for $a>2$. Hence, we can directly apply Proposition 4 in
berkes2011split, obtaining that, for all $1\leq j\leq d$, there
exist constants $c_{j}<\infty $ such that
\begin{equation}
\left\vert \sum_{t=1}^{m}x_{j,t}{\Greekmath 010F} _{t}\right\vert _{2}^{2}\leq c_{j}m.
\end{equation}
Putting ((ref)) in ((ref)), we finally obtain that $
E\left\Vert \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert ^{2}\leq
c_{0}dm$, whence the desired result follows from Markov inequality.
lemmaWe assume that Assumptions (ref) and (ref) hold. Then it holds that
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}_{1})\right\Vert =O_{P}(d^{1/2}),
\end{equation*}
for all ${\Greekmath 0110} _{3}>1/2,$ where $\mathbf{c}
_{1}=(1,E(x_{2,0}),...,E(x_{d,0}))^{\prime }$.
proofIt holds that
\begin{align*}
&P\left( \max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}_{1})\right\Vert \geq
xd^{1/2}\right) \\
\leq &P\left( \max_{0\leq \ell \leq \left\lceil \ln T_{m}\right\rceil
}\max_{\exp \left( \ell \right) \leq k\leq \exp \left( \ell +1\right) }\frac{
1}{k^{{\Greekmath 0110} _{3}}}\left\Vert \sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}
_{1})\right\Vert \geq xd^{1/2}\right) \\
\leq &\sum_{\ell =0}^{\left\lceil \ln T_{m}\right\rceil }P\left( \max_{\exp
\left( \ell \right) \leq k\leq \exp \left( \ell +1\right) }\frac{1}{k^{{\Greekmath 0110}
_{3}}}\left\Vert \sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}
_{1})\right\Vert \geq xd^{1/2}\right) \\
\leq &\sum_{\ell =0}^{\left\lceil \ln T_{m}\right\rceil }P\left( \max_{\exp
\left( \ell \right) \leq k\leq \exp \left( \ell +1\right) }\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}_{1})\right\Vert \geq
xd^{1/2}\exp \left( {\Greekmath 0110} _{3}\ell \right) \right) \\
\leq &c_{0}x^{-p}d^{-p/2}\sum_{\ell =0}^{\left\lceil \ln T_{m}\right\rceil
}\exp \left( -p{\Greekmath 0110} _{3}\ell \right) E\left( \max_{\exp \left( \ell \right)
\leq k\leq \exp \left( \ell +1\right) }\left\Vert \sum_{t=m+1}^{m+k}(\mathbf{
x}_{t}-\mathbf{c}_{1})\right\Vert ^{p}\right)
\end{align*}
for any $2\leq p\leq 4$. Further, we have
\begin{align*}
&E\left( \max_{\exp \left( \ell \right) \leq k\leq \exp \left( \ell
+1\right) }\left\Vert \sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}
_{1})\right\Vert ^{p}\right) \\
\leq &d^{p/2-1}\sum_{j=2}^{d}E\left( \max_{\exp \left( \ell \right) \leq
k\leq \exp \left( \ell +1\right) }\left\vert
\sum_{t=m+1}^{m+k}(x_{j,t}-Ex_{j,0})\right\vert ^{p}\right) \\
\leq &d^{p/2-1}\sum_{j=2}^{d}E\left( \max_{\exp \left( \ell \right) \leq
k\leq \exp \left( \ell +1\right) }\left\vert
\sum_{t=m+1}^{m+k}(x_{j,t}-Ex_{j,0})\right\vert ^{p}\right) .
\end{align*}
Recall that $x_{j,t}-Ex_{j,0}$ is a zero mean, $L_{4}$-decomposable
Bernoulli shift which satisfies the assumptions of Proposition 4 in
berkes2011split; using Theorem 1 in moricz1976moment, it
follows that
\begin{equation*}
E\left( \max_{1\leq k\leq \exp \left( \ell \right) }\left\vert
\sum_{t=m+1}^{m+k}(x_{j,t}-Ex_{j,0})\right\vert ^{p}\right) \leq c_{j}\exp
\left( \frac{p}{2}\ell \right) .
\end{equation*}
Hence, putting all together it follows that
\begin{equation*}
P\left( \max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\mathbf{c}_{1})\right\Vert \geq
xd^{1/2}\right) \leq c_{0}x^{-p}\sum_{\ell =0}^{\ln T_{m}}\exp \left( p\ell
\left( -{\Greekmath 0110} _{3}+\frac{1}{2}\right) \right) \leq c_{1},
\end{equation*}
whenever ${\Greekmath 0110} _{3}>1/2$, which proves the desired result.
lemmaWe assume that Assumptions (ref) and (ref) hold. Then it holds that
\begin{equation*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\left( \mathbf{x}_{t}\mathbf{x}
_{t}^{\prime }-\mathbf{C}\right) \right\Vert _{F}=O_{P}\left( \frac{d}{\sqrt{
m}}\right)
\end{equation*}
proofWe begin by showing that, for all $2\leq h,j\leq d$, $x_{h,t}x_{j,t}$ is an $
L_{2}$-decomposable Bernoulli shift with $a>2$. Indeed, consider the coupling
\begin{equation*}
\widetilde{x}_{j,t,t}=g_{j}^{x}\left( {\Greekmath 0111} _{j,t}^{x},...,{\Greekmath 0111} _{j,1}^{x},
\widetilde{{\Greekmath 0111} }_{j,0,t,t}^{x},\widetilde{{\Greekmath 0111} }_{j,-1,t,t}^{x},...\right) ,
\end{equation*}
where $\left\{ \widetilde{{\Greekmath 0111} }_{j,0,t,t}^{x},\widetilde{{\Greekmath 0111} }
_{j,-1,t,t}^{x},...\right\} $ are independent copies of $\left\{ {\Greekmath 0111}
_{j,0}^{x},{\Greekmath 0111} _{j,-1}^{x},...\right\} $, and similarly $\left\{ \widetilde{
{\Greekmath 0111} }_{j,0,t,t}^{{\Greekmath 010F} },\widetilde{{\Greekmath 0111} }_{j,-1,t,t}^{{\Greekmath 010F}
},...\right\} $, $2\leq j\leq d$. It holds that
\begin{align*}
x_{h,t}x_{j,t} =&\left( x_{h,t}\pm \widetilde{x}_{h,t,t}\right) \left(
x_{j,t}\pm \widetilde{x}_{j,t,t}\right) \\
=&\widetilde{z}_{j,h,t,t}+\widetilde{x}_{j,t,t}\left( x_{h,t}-\widetilde{x}
_{h,t,t}\right) +\widetilde{x}_{h,t,t}\left( x_{j,t}-\widetilde{x}
_{j,t,t}\right) +\left( x_{j,t}-\widetilde{x}_{j,t,t}\right) \left( x_{h,t}-
\widetilde{x}_{h,t,t}\right) ,
\end{align*}
where $\widetilde{z}_{j,h,t,t}=\widetilde{x}_{j,t,t}\widetilde{x}_{h,t,t}$
is the coupled version of $z_{j,h,t}=x_{h,t}x_{j,t}$. Hence it follows that
\begin{align*}
\left\vert z_{j,t}-\widetilde{z}_{j,h,t,t}\right\vert _{2} \leq &\left\vert
\widetilde{x}_{j,t,t}\left( x_{h,t}-\widetilde{x}_{h,t,t}\right) \right\vert
_{2}+\left\vert \widetilde{x}_{h,t,t}\left( x_{j,t}-\widetilde{x}
_{j,t,t}\right) \right\vert _{2}+\left\vert \left( x_{j,t}-\widetilde{x}
_{j,t,t}\right) \left( x_{h,t}-\widetilde{x}_{h,t,t}\right) \right\vert _{2}
\\
\leq &\left\vert \widetilde{x}_{j,t,t}\right\vert _{4}\left\vert x_{h,t}-
\widetilde{x}_{h,t,t}\right\vert _{4}+\left\vert x_{j,t}-\widetilde{x}
_{j,t,t}\right\vert _{4}\left\vert \widetilde{x}_{h,t,t}\right\vert
_{4}+\left\vert x_{j,t}-\widetilde{x}_{j,t,t}\right\vert _{4}\left\vert
x_{h,t}-\widetilde{x}_{h,t,t}\right\vert _{4}
\end{align*}
having used Minkowski's inequality in the first line, and the
Cauchy-Schwartz inequality in the second one. Assumption (ref)
(ii) now immediately yields that there exists a constant $
0<c_{0}<\infty $ such that
\begin{equation*}
\left\vert z_{j,h,t}-\widetilde{z}_{j,h,t,t}\right\vert _{2}\leq c_{0}t^{-a}.
\end{equation*}
We now note that
\begin{equation*}
\left\Vert \sum_{t=1}^{m}\left( \mathbf{x}_{t}\mathbf{x}_{t}^{\prime }-
\mathbf{C}\right) \right\Vert _{F}^{2}=\sum_{j,h=1}^{d}\left\vert
\sum_{t=1}^{m}\left( x_{h,t}x_{j,t}-\mathbf{C}_{h,j}\right) \right\vert ^{2}.
\end{equation*}
Using Proposition 4 in berkes2011split, it follows that
\begin{equation*}
\left\vert \sum_{t=1}^{m}\left( x_{h,t}x_{j,t}-\mathbf{C}_{h,j}\right)
\right\vert _{2}^{2}\leq c_{hj}m,
\end{equation*}
where the constants $c_{hj}$ are finite. Hence the desired result
immediately follows from Markov inequality.
lemmaWe assume that Assumptions (ref)-(ref)
, and $d=O\left( m^{1/4}\right) $, hold. Then it holds that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
\sum_{t=m+1}^{m+k}\widehat{{\Greekmath 010F} }_{t}\right\vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \left( \frac{k}{k+m}\right)
^{{\Greekmath 0111} }} \\
=&r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
\sum_{t=m+1}^{m+k}{\Greekmath 010F} _{t}-\frac{k}{m}
\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right\vert }{m^{1/2}\left( 1+\frac{k
}{m}\right) \left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }}+o_{P}(1).
\end{align*}
Note that
align*[align* omitted — 541 chars of source]
We begin by showing that
equation*[equation* omitted — 503 chars of source]
Indeed
align*[align* omitted — 989 chars of source]
By stationarity implied by Assumption (ref), and standard
arguments, it follows immediately that $\left\Vert \sum_{t=m+1}^{m+k}\mathbf{
x}_{t}\right\Vert =O_{P}\left( d^{1/2}k\right) $. Further, note that, using
Taylor's expansion
align*[align* omitted — 1,207 chars of source]
Hence it holds that
align*[align* omitted — 462 chars of source]
and
equation*[equation* omitted — 353 chars of source]
using Lemma (ref) and seeing as $s_{\max }\left( \mathbf{C}
^{-1}\right) =s_{\min }^{-1}\left( \mathbf{C}\right) $ and, by Assumption
(ref)(iii), $s_{\min }\left( \mathbf{C}\right) >0$. By a
similar logic
equation*[equation* omitted — 345 chars of source]
Thus it finally follows that
equation[equation omitted — 200 chars of source]
Finally, using Lemma (ref) and ((ref))
align*[align* omitted — 951 chars of source]
We note that, depending on the values of ${\Greekmath 0111} $, there are two bounds. If $
{\Greekmath 0111} \leq 1$, then $\max_{a_{m}\leq k\leq T_{m}}\left( k/\left( m+k\right)
\right) ^{1-{\Greekmath 0111} }$ is bounded; conversely, if ${\Greekmath 0111} >1$, the maximum is
attained at $\left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 0111} }$. Hence,
putting all together, it finally holds that
align*[align* omitted — 622 chars of source]
on account of the fact that $d=O\left( m^{1/4}\right) $. Therefore it
follows that
align*[align* omitted — 802 chars of source]
We now show that
equation*[equation* omitted — 433 chars of source]
Indeed
align*[align* omitted — 854 chars of source]
Using Lemmas (ref)-(ref), this entails that
align*[align* omitted — 937 chars of source]
By virtue of Lemma (ref), we can choose $1/2<{\Greekmath 0110} _{3}<\min
\{1,{\Greekmath 0111} \}$ in the above, so that
align*[align* omitted — 716 chars of source]
In order to bound $\left\Vert \mathbf{C}^{-1}\right\Vert _{F}$, we note that
equation*[equation* omitted — 167 chars of source]
where $\left\Vert \mathbf{C}^{-1}\right\Vert _{2}$ is the $L_{2}$-norm of a
matrix. Again by Assumption (ref)(iii), $s_{\min }\left(
\mathbf{C}\right) >0$, which entails that there exists a finite constant $
c_{0}$ such that
equation[equation omitted — 92 chars of source]
Hence it holds that
equation*[equation* omitted — 505 chars of source]
under the constraint $d=O\left( m^{1/4}\right) $, this finally entails that
align*[align* omitted — 763 chars of source]
By construction, $\mathbf{c}_{1}^{\prime }\mathbf{C}^{-1}=\left(
1,0,...,0\right) ^{\prime }$, and therefore it finally follows that
equation*[equation* omitted — 537 chars of source]
The desired result now follows from putting all together.
lemmaWe assume that Assumptions (ref)(i) and
(ref) hold. Then, on a suitably enlarged probability space, it is
possible to define two independent standard Wiener processes $\left\{
W_{1,m}\left( k\right) ,k\geq 1\right\} $ and $\left\{ W_{2,m}\left(
k\right) ,k\geq 1\right\} $ such that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{1}{{\Greekmath 011B} }\frac{
\left\vert \sum_{t=m+1}^{m+k}{\Greekmath 010F} _{t}-\frac{k
}{m}\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right\vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \left( \frac{k}{k+m}\right)
^{{\Greekmath 0111} }} \\
=&r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
W_{1,m}\left( k\right) -\frac{k}{m}W_{2,m}\left( m\right)
\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }}+o_{P}\left( 1\right) .
\end{align*}
Henceforth, we set ${\Greekmath 011B} =1$ for simplicity and without loss of
generality. Standard arguments entail that
align*[align* omitted — 1,395 chars of source]
Using ((ref)) in Lemma (ref), it follows that
align*[align* omitted — 1,093 chars of source]
Similarly, using ((ref)) in Lemma (ref)
align*[align* omitted — 893 chars of source]
Hence, after some algebra, it can be shown that
equation*[equation* omitted — 434 chars of source]
The desired result now follows.
\setcounter{equation}{0} \setcounter{lemma}{0} \setcounter{theorem}{0}
Proofs
proof[Proof of Theorem (ref)]
Lemmas (ref)-(ref), put together, entail that
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{1}{{\Greekmath 011B} }\frac{
\left\vert \sum_{t=m+1}^{m+k}\widehat{{\Greekmath 010F} }_{t}\right\vert
}{m^{1/2}\left( 1+\frac{k}{m}\right) \left( \frac{k
}{k+m}\right) ^{{\Greekmath 0111} }} \\
=& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
W_{1,m}\left( k\right) -\frac{k}{m}W_{2,m}\left( m\right)
\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }}+o_{P}\left( 1\right) .
\end{align*}
As in the other proofs, we will set ${\Greekmath 011B} =1$ for simplicity and without
loss of generality. We begin by noting that the distribution of $
W_{1,m}\left( \cdot \right) $ and $W_{2,m}\left( \cdot \right) $ does not
depend on $m$, so that
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert
W_{1,m}\left( k\right) -\frac{k}{m}W_{2,m}\left( m\right)
\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{
\left\vert W_{1}\left( k\right) -\frac{k}{m}W_{2}\left(
m\right) \right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{
\left\vert W_{1}\left( \frac{k}{m}\right) -\frac{k
}{m}W_{2}\left( 1\right) \right\vert }{\left( 1+\frac{k}{m}
\right) \left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }}.
\end{align*}
By standard arguments, it is easy to see that
\begin{equation}
W_{1}\left( t\right) -tW_{2}\left( 1\right) \overset{\mathcal{D}}{=}\left(
1+t\right) W\left( \frac{t}{1+t}\right) ,
\end{equation}
for all $t$, where $W\left( \cdot \right) $ is a standard Wiener.\footnote{
Indeed, this can be verified readily by noting that both processes $
W_{1}\left( s\right) $ and $W_{2}\left( s\right) $ have mean zero, are
Gaussian, and have covariance kernel given by
\begin{align*}
& E\left( W_{1}\left( t\right) -tW_{2}\left( 1\right) \right) \left(
W_{1}\left( s\right) -sW_{2}\left( 1\right) \right) \\
=& E\left( W_{1}\left( t\right) W_{1}\left( s\right) \right) -stE\left(
W_{2}\left( 1\right) \right) ^{2} \\
=& s\wedge t-st,
\end{align*}
and
\begin{align*}
& E\left( \left( 1+t\right) W\left( \frac{t}{1+t}\right) \left( 1+s\right)
W\left( \frac{s}{1+s}\right) \right) \\
=& \left( 1+t\right) \left( 1+s\right) \left( \frac{t}{1+t}\wedge \frac{s}{
1+s}\right) ,
\end{align*}
respectively - it is now not hard to see that the two covariance kernels are
the same.} Hence we will study
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{\left\vert \left( 1+
\frac{k}{m}\right) W\left( \frac{\frac{k}{m}}{1+
\frac{k}{m}}\right) \right\vert }{\left( 1+\frac{k}{m}\right)
\left( \frac{k}{k+m}\right) ^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}/m\leq t\leq T_{m}/m}
\frac{\left\vert W\left( \frac{t}{1+t}\right) \right\vert }{
\left( \frac{t}{1+t}\right) ^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}/\left( m+a_{m}\right)
\leq s\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left( s\right)
\right\vert }{s^{{\Greekmath 0111} }}.
\end{align*}
Define
\begin{equation*}
u=s\frac{a_{m}+m}{a_{m}};
\end{equation*}
it follows that
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}/\left( m+a_{m}\right) \leq s\leq T_{m}/\left(
m+T_{m}\right) }\frac{\left\vert W\left( s\right) \right\vert }{s^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{1\leq u\leq T_{m}\left(
m+a_{m}\right) /\left( a_{m}\left( m+T_{m}\right) \right) }\frac{\left\vert W
\left( u\frac{a_{m}}{a_{m}+m}\right) \right\vert }{
\left( u\frac{a_{m}}{a_{m}+m}\right) ^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\left( \frac{a_{m}}{a_{m}+m}
\right) ^{1/2-{\Greekmath 0111} }\max_{1\leq u\leq T_{m}\left( m+a_{m}\right) /\left(
a_{m}\left( m+T_{m}\right) \right) }\frac{\left\vert W\left( u\right)
\right\vert }{u^{{\Greekmath 0111} }} \\
& \overset{\mathcal{D}}{=}\max_{1\leq u\leq T_{m}\left( m+a_{m}\right)
/\left( a_{m}\left( m+T_{m}\right) \right) }\frac{\left\vert W\left(
u\right) \right\vert }{u^{{\Greekmath 0111} }},
\end{align*}
having used the scale transformation of the Wiener process in the third
passage, and the definition of $r_{m}$ in the last one. Recalling ((ref)
), by elementary arguments it holds that
\begin{equation*}
\lim_{m\rightarrow \infty }\frac{T_{m}\left( m+a_{m}\right) }{a_{m}\left(
m+T_{m}\right) }\rightarrow \infty ,
\end{equation*}
whence, as $m\rightarrow \infty $, it follows that, by continuity
\begin{equation*}
\max_{1\leq u\leq T_{m}\left( m+a_{m}\right) /\left( a_{m}\left(
m+T_{m}\right) \right) }\frac{\left\vert W\left( u\right) \right\vert }{
u^{{\Greekmath 0111} }}\overset{a.s.}{\rightarrow }\sup_{1\leq u<\infty }\frac{\left\vert
W(u)\right\vert }{u^{{\Greekmath 0111} }}.
\end{equation*}
As a final remark, we reconsider the proof, showing that the limiting
distribution is determined only by the observations very close to $a_{m}$.
To show this, we divide into two intervals the domain where the maximum is
taken defining
\begin{align}
& \Theta _{m,1}=r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}/\left( m+a_{m}\right) \leq
s<\left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }}\frac{\left\vert
W\left( s\right) \right\vert }{s^{{\Greekmath 0111} }}, \\
& \Theta _{m,2}=r_{m}^{{\Greekmath 0111} -1/2}\max_{\left( a_{m}/\left( m+a_{m}\right)
\right) ^{1-{\Greekmath 010F} }\leq s<T_{m}/\left( m+T_{m}\right) }\frac{\left\vert
W\left( s\right) \right\vert }{s^{{\Greekmath 0111} }}.
\end{align}
for some ${\Greekmath 010F} >0$ such that $\left( a_{m}/\left( m+a_{m}\right) \right)
^{1-{\Greekmath 010F} }=o\left( T_{m}/\left( m+T_{m}\right) \right) $. We begin by
studying $\Theta _{m,2}$. Repeatedly using the scale transformation for
Wiener process, we have
\begin{align*}
& \Theta _{m,2}\overset{\mathcal{D}}{=}r_{m}^{{\Greekmath 0111} -1/2}\max_{\left(
a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }\leq s\leq T_{m}/\left(
m+T_{m}\right) }\frac{s^{1/2}}{s^{1/2}}\frac{\left\vert W\left( s\right)
\right\vert }{s^{{\Greekmath 0111} }} \\
& \leq r_{m}^{{\Greekmath 0111} -1/2}\max_{\left( a_{m}/\left( m+a_{m}\right) \right)
^{1-{\Greekmath 010F} }\leq s\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert
W\left( s\right) \right\vert }{s^{1/2}}\max_{\left( a_{m}/\left(
m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }\leq s\leq T_{m}/\left( m+T_{m}\right)
}s^{1/2-{\Greekmath 0111} } \\
& \leq \left( \left( \frac{a_{m}}{a_{m}+m}\right) ^{1/2}\right) ^{{\Greekmath 0111}
-1/2}\max_{\left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }\leq
s\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left( s\right)
\right\vert }{s^{1/2}}.
\end{align*}
We note that
\begin{equation*}
\max_{\left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }\leq s\leq
T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left( s\right) \right\vert }{
s^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq u\leq \left( T_{m}\left(
m+a_{m}\right) ^{1-{\Greekmath 010F} }\right) /\left( \left( m+T_{m}\right)
a_{m}^{1-{\Greekmath 010F} }\right) }\frac{\left\vert W\left( u\right) \right\vert }{
u^{1/2}},
\end{equation*}
so that the Law of the Iterated Logarithm entails that
\begin{equation*}
\max_{\left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }\leq s\leq
T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left( s\right) \right\vert }{
s^{1/2}}=O_{P}\left( \sqrt{\ln \ln \frac{T_{m}\left( m+a_{m}\right)
^{1-{\Greekmath 010F} }}{\left( m+T_{m}\right) a_{m}^{1-{\Greekmath 010F} }}}\right)
=O_{P}\left( \sqrt{\ln \ln \left( \frac{m}{a_{m}}\right) ^{1-{\Greekmath 010F} }}
\right) .
\end{equation*}
Hence, recalling ((ref))
\begin{equation*}
\Theta _{m,2}=O_{P}\left( \left( \left( \frac{a_{m}}{a_{m}+m}\right)
^{1/2}\right) ^{{\Greekmath 0111} -1/2}\sqrt{\ln \ln \left( \frac{m}{a_{m}}\right)
^{1-{\Greekmath 010F} }}\right) =o_{P}\left( 1\right) .
\end{equation*}
Using this result in conjunction with Lemmas (ref)-(ref) it
follows that, as $m\rightarrow \infty $
\begin{equation*}
P\left( r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq T_{m}}\frac{1}{{\Greekmath 011B} }\frac{
\left\vert \sum_{t=m+1}^{m+k}\widehat{{\Greekmath 010F} }_{t}\right\vert
}{m^{1/2}\left( 1+\frac{k}{m}\right) \left( \frac{k
}{k+m}\right) ^{{\Greekmath 0111} }}=\Theta _{m,1}\right) =1.
\end{equation*}
We now conclude our remarks by studying $\Theta _{m,1}$ in ((ref)).
Let
\begin{equation*}
u=s\frac{a_{m}+m}{a_{m}}.
\end{equation*}
Then it holds that
\begin{align*}
\Theta _{m,1}=& r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}/\left( m+a_{m}\right) \leq
s\leq \left( a_{m}/\left( m+a_{m}\right) \right) ^{1-{\Greekmath 010F} }}\frac{
\left\vert W\left( s\right) \right\vert }{s^{{\Greekmath 0111} }} \\
\overset{\mathcal{D}}{=}& r_{m}^{{\Greekmath 0111} -1/2}\max_{1\leq u\leq \left( \left(
m+a_{m}\right) /a_{m}\right) ^{{\Greekmath 010F} }}\frac{\left\vert W
\left( u\frac{a_{m}}{a_{m}+m}\right) \right\vert }{\left( u
\frac{a_{m}}{a_{m}+m}\right) ^{{\Greekmath 0111} }} \\
\overset{\mathcal{D}}{=}& r_{m}^{{\Greekmath 0111} -1/2}\left( \frac{a_{m}}{a_{m}+m}
\right) ^{1/2-{\Greekmath 0111} }\max_{1\leq u\leq \left( \left( m+a_{m}\right)
/a_{m}\right) ^{{\Greekmath 010F} }}\frac{\left\vert W\left( u\right) \right\vert }{
u^{{\Greekmath 0111} }} \\
\overset{\mathcal{D}}{=} & \max_{1\leq u\leq \left( \left( m+a_{m}\right)
/a_{m}\right) ^{{\Greekmath 010F} }}\frac{\left\vert W\left( u\right) \right\vert }{
u^{{\Greekmath 0111} }},
\end{align*}
and
\begin{equation*}
\Theta _{m,1}\overset{\mathcal{D}}{=}\max_{1\leq u\leq \left( \left(
m+a_{m}\right) /a_{m}\right) ^{{\Greekmath 010F} }}\frac{\left\vert W\left( u\right)
\right\vert }{u^{{\Greekmath 0111} }}\overset{a.s.}{\rightarrow }\sup_{1\leq u<\infty }
\frac{\left\vert W(u)\right\vert }{u^{{\Greekmath 0111} }}.
\end{equation*}
proof[Proof of Lemma (ref)]
The proof of the lemma is standard, and we only report its main passages,
mainly to highlight the impact of $d$. Let ${\Greekmath 010D} _{j}=E\left( {\Greekmath 010F}
_{0}{\Greekmath 010F} _{j}\right) $. By Assumption (ref), it is easy to see
that
\begin{equation*}
{\Greekmath 011B} ^{2}=\lim_{m\rightarrow \infty }\sum_{t,s=1}^{m}E\left( {\Greekmath 010F}
_{t}{\Greekmath 010F} _{s}\right) ={\Greekmath 010D} _{0}+2\sum_{j=1}^{\infty }{\Greekmath 010D} _{j}.
\end{equation*}
We begin by showing that the ${\Greekmath 010D} _{j}$s satisfy
\begin{equation}
\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010D} _{j}\right\vert <\infty .
\end{equation}
Indeed, let ${\Greekmath 010F} _{j}=h\left( {\Greekmath 0111} _{j},{\Greekmath 0111} _{j-1},...\right) $, and
define ${\Greekmath 010F} _{j,j}=h\left( {\Greekmath 0111} _{j},{\Greekmath 0111} _{j-1},...,{\Greekmath 0111} _{1},
\widetilde{{\Greekmath 0111} }_{0,j,j},\widetilde{{\Greekmath 0111} }_{-1,j,j},...\right) $ where, as
per Definition (ref), $\left\{ \widetilde{{\Greekmath 0111} }_{t,j,j},-\infty
<t,j<\infty \right\} $ i.i.d. copies of ${\Greekmath 0111} _{0}$ independent of $
\left\{ {\Greekmath 0111} _{t},-\infty <t<\infty \right\} $. By construction, ${\Greekmath 010F}
_{0}$ and ${\Greekmath 010F} _{j,j}$ are independent and have mean zero, so that $
E\left( {\Greekmath 010F} _{0}{\Greekmath 010F} _{j,j}\right) =0$ and
\begin{gather*}
\sum_{j=1}^{\infty }j\left\vert E\left( {\Greekmath 010F} _{0}{\Greekmath 010F} _{j}\right)
\right\vert =\sum_{j=1}^{\infty }j\left\vert E\left( {\Greekmath 010F} _{0}\left(
{\Greekmath 010F} _{j}-{\Greekmath 010F} _{j,j}\right) \right) \right\vert \leq
\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010F} _{0}\right\vert _{2}\left\vert
{\Greekmath 010F} _{j}-{\Greekmath 010F} _{j,j}\right\vert _{2} \\
\leq c_{0}\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010F} _{j}-{\Greekmath 010F}
_{j,j}\right\vert _{2}\leq c_{0}\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010F}
_{j}-{\Greekmath 010F} _{j,j}\right\vert _{4}\leq c_{0}\sum_{j=1}^{\infty
}j^{1-a}\leq c_{1},
\end{gather*}
having used: the Cauchy-Schwartz inequality in the third passage; the fact
that, by Assumption (ref)(i) $\left\vert {\Greekmath 010F}
_{0}\right\vert _{2}<\infty $ in the fourth passage; the $L_{p}$-norm
inequality in the fifth passage; and Assumption (ref)(i)
in the last one, recalling that $a>2$. We now write
\begin{align*}
\widehat{{\Greekmath 011B} }_{m}^{2}-{\Greekmath 011B} ^{2} =&\left( \widehat{{\Greekmath 010D} }_{0}-{\Greekmath 010D}
_{0}\right) +2\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \widehat{
{\Greekmath 010D} }_{j}-{\Greekmath 010D} _{j}\right) +2\sum_{j=1}^{H}\frac{j}{H+1}{\Greekmath 010D}
_{j}-2\sum_{j=H+1}^{\infty }{\Greekmath 010D} _{j} \\
=&I+II+III+IV.
\end{align*}
By ((ref)), it follows that
\begin{equation*}
\frac{1}{2}III\leq \frac{1}{H+1}\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010D}
_{j}\right\vert \leq c_{0}\frac{1}{H+1};
\end{equation*}
similarly
\begin{equation*}
\frac{1}{2}IV\leq \sum_{j=H+1}^{\infty }\frac{j}{H+1}\left\vert {\Greekmath 010D}
_{j}\right\vert \leq \frac{1}{H+1}\sum_{j=1}^{\infty }j\left\vert {\Greekmath 010D}
_{j}\right\vert \leq c_{0}\frac{1}{H+1},
\end{equation*}
so that $III+IV=O\left( H^{-1}\right) $. We now turn to $I$ and $II$\ noting
that
\begin{align*}
&\widehat{{\Greekmath 010D} }_{j}-{\Greekmath 010D} _{j}=\frac{1}{m}\sum_{t=j+1}^{m}\widehat{
{\Greekmath 010F} }_{t}\widehat{{\Greekmath 010F} }_{t-j}-E\left( {\Greekmath 010F} _{0}{\Greekmath 010F}
_{j}\right) \\
=&\frac{1}{m}\sum_{t=j+1}^{m}\left( {\Greekmath 010F} _{t}{\Greekmath 010F} _{t-j}-E\left(
{\Greekmath 010F} _{0}{\Greekmath 010F} _{j}\right) \right) +\left( {\Greekmath 010C} -\widehat{{\Greekmath 010C} }
_{m}\right) ^{\prime }\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t-j} \\
&+\left( {\Greekmath 010C} -\widehat{{\Greekmath 010C} }_{m}\right) ^{\prime }\frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t-j}{\Greekmath 010F} _{t}+\left( \widehat{{\Greekmath 010C} }
_{m}-{\Greekmath 010C} \right) ^{\prime }\left( \frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}
\mathbf{x}_{t-j}^{\prime }\right) \left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right) ,
\end{align*}
whence
\begin{align*}
&\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \widehat{{\Greekmath 010D} }
_{j}-{\Greekmath 010D} _{j}\right) \\
=&\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\left( {\Greekmath 010F} _{t}{\Greekmath 010F} _{t-j}-E\left( {\Greekmath 010F}
_{0}{\Greekmath 010F} _{j}\right) \right) \right) \\
&+\left( {\Greekmath 010C} -\widehat{{\Greekmath 010C} }_{m}\right) ^{\prime }\left[
\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t-j}\right) \right] \\
&+\left( {\Greekmath 010C} -\widehat{{\Greekmath 010C} }_{m}\right) ^{\prime }\left[
\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t-j}{\Greekmath 010F} _{t}\right) \right] \\
&+\left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right) ^{\prime }\left[
\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime }\right) \right]
\left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right) \\
=&II_{a}+II_{b}+II_{c}+II_{d}.
\end{align*}
It holds that
\begin{align*}
&E\left( \sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\left( {\Greekmath 010F} _{t}{\Greekmath 010F} _{t-j}-E\left( {\Greekmath 010F}
_{0}{\Greekmath 010F} _{j}\right) \right) \right) \right) ^{2} \\
\leq &c_{0}H\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) ^{2}E\left( \frac{1}{
m}\sum_{t=j+1}^{m}\left( {\Greekmath 010F} _{t}{\Greekmath 010F} _{t-j}-E\left( {\Greekmath 010F}
_{0}{\Greekmath 010F} _{j}\right) \right) \right) ^{2} \\
\leq &c_{0}H\sum_{j=1}^{H}E\left( \frac{1}{m}\sum_{t=j+1}^{m}\left( {\Greekmath 010F}
_{t}{\Greekmath 010F} _{t-j}-E\left( {\Greekmath 010F} _{0}{\Greekmath 010F} _{j}\right) \right)
\right) ^{2}.
\end{align*}
By Assumption (ref)(i) and the same logic as in the proof
of Lemma (ref), it follows that ${\Greekmath 010F} _{t}{\Greekmath 010F} _{t-j}$ is
an $L_{2}$-decomposable Bernoulli shift for all $j$; hence, using
Proposition 4 in berkes2011split, it follows immediately that
\begin{equation*}
H\sum_{j=1}^{H}E\left( \frac{1}{m}\sum_{t=j+1}^{m}\left( {\Greekmath 010F}
_{t}{\Greekmath 010F} _{t-j}-E\left( {\Greekmath 010F} _{0}{\Greekmath 010F} _{j}\right) \right)
\right) ^{2}\leq c_{0}\frac{H^{2}}{m},
\end{equation*}
which entails that $II_{a}=O_{P}\left( Hm^{-1/2}\right) $. Standard
passages, using Assumption (ref)(iii), entail that
\begin{equation}
\left\Vert \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right\Vert =O_{P}\left( \sqrt{\frac{d
}{m}}\right) ;
\end{equation}
further
\begin{align*}
&E\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t-j}\right) \\
\leq &E\sum_{j=1}^{H}\left\Vert \frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}
_{t}{\Greekmath 010F} _{t-j}\right\Vert \leq c_{0}\frac{1}{m}\sum_{j=1}^{H}
\sum_{t=j+1}^{m}\left( E\left\Vert \mathbf{x}_{t}\right\Vert ^{2}\right)
^{1/2}\left( E\left\vert {\Greekmath 010F} _{t-j}\right\vert ^{2}\right) ^{1/2}\leq
c_{0}H;
\end{align*}
this bound is not the sharpest possible (one could assume $E\left( \mathbf{x}
_{t}{\Greekmath 010F} _{t-j}\right) =0$ and then use the Markov inequality), but it
suffices for our purposes. Using ((ref)), this entails that both $
II_{b}$ and $II_{c}$ are $O_{P}\left( d^{1/2}Hm^{-1/2}\right) $. Finally
\begin{align*}
&\left\vert \left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right) ^{\prime }\left[
\sum_{j=1}^{H}\left( 1-\frac{j}{H+1}\right) \left( \frac{1}{m}
\sum_{t=j+1}^{m}\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime }\right) \right]
\left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right) \right\vert \\
\leq &\left\Vert \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} \right\Vert ^{2}\frac{1}{m}
\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left\vert \mathbf{x}_{t}\mathbf{x}
_{t-j}^{\prime }\right\vert ,
\end{align*}
and since $\left\vert \mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime }\right\vert
\leq \left\Vert \mathbf{x}_{t}\right\Vert \left\Vert \mathbf{x}
_{t-j}\right\Vert $, using the Cauchy-Schwartz inequality and the
stationarity of $\mathbf{x}_{t}$
\begin{equation*}
E\frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left\vert \mathbf{x}_{t}\mathbf{x}
_{t-j}^{\prime }\right\vert \leq \frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}
\left( E\left\Vert \mathbf{x}_{t}\right\Vert ^{2}\right) ^{1/2}\left(
E\left\Vert \mathbf{x}_{t-j}\right\Vert ^{2}\right) ^{1/2}\leq c_{0}H.
\end{equation*}
Thus, again using ((ref)), it follows that $II_{d}=O_{P}\left(
Hdm^{-1}\right) $. The final result now follows from putting all together.
proof[Proof of Theorem (ref)]
The proof is related to that of Theorem 2.2 in horvath2004monitoring
, and we only report the main passages for the sake of a concise discussion.
Standard algebra entails that
\begin{equation*}
\sum_{t=m+1}^{m+\widetilde{k}}\widehat{{\Greekmath 010F} }_{t}=\sum_{t=m+1}^{m+
\widetilde{k}}{\Greekmath 010F} _{t}-\left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} _{0}\right)
^{\prime }\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}+\Delta _{m}^{\prime
}\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}I\left( t>m+k^{\ast }\right) .
\end{equation*}
The proof of Theorem (ref) immediately implies that
\begin{equation*}
r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+\widetilde{k}
}{\Greekmath 010F} _{t}-\left( \widehat{{\Greekmath 010C} }_{m}-{\Greekmath 010C} _{0}\right) ^{\prime }
\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right\vert }{
g_{{\Greekmath 0111} }\left( m;\widetilde{k}\right) }=O_{P}\left( 1\right) .
\end{equation*}
Also, using Lemma (ref), it is not hard to see that
\begin{align*}
& \Delta _{m}^{\prime }\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}I\left(
t>k^{\ast }\right) \\
=& \left( \widetilde{k}-k^{\ast }\right) \mathbf{c}_{1}^{\prime }\Delta _{m}+
\mathbf{c}_{1}^{\prime }\Delta _{m}O_{P}\left( \left( \widetilde{k}-k^{\ast
}\right) ^{1/2}\right) =\mathbf{c}_{1}^{\prime }\Delta _{m}\left(
a_{m}+O_{P}\left( a_{m}^{1/2}\right) \right) .
\end{align*}
Recall that, by assumption, it holds that $\left\vert \mathbf{c}_{1}^{\prime
}\Delta _{m}\right\vert >0$. Under condition (i), let $\widetilde{k}
=k^{\ast }+a_{m}$. It follows that
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \Delta _{m}^{\prime }
\sum_{t=m+k^{\ast }+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right\vert }{
m^{1/2}\left( 1+\frac{\widetilde{k}}{m}\right)
\left( \frac{\widetilde{k}}{m+\widetilde{k}}\right) ^{{\Greekmath 0111} }} \\
=& r_{m}^{{\Greekmath 0111} -1/2}\frac{a_{m}}{m^{1/2}\left( 1+\frac{
\widetilde{k}}{m}\right) \left( \frac{\widetilde{k}}{m+
\widetilde{k}}\right) ^{{\Greekmath 0111} }}\mathbf{c}_{1}^{\prime }\Delta
_{m}+O_{P}\left( \mathbf{c}_{1}^{\prime }\Delta _{m}\right) \\
=& a_{m}^{1/2}\mathbf{c}_{1}^{\prime }\Delta _{m}+O_{P}\left( \mathbf{c}
_{1}^{\prime }\Delta _{m}\right) +o\left( a_{m}^{1/2}\mathbf{c}_{1}^{\prime
}\Delta _{m}\right) ,
\end{align*}
whence the theorem follows immediately if ((ref)) holds. Under
condition (ii), let $\widetilde{k}=\left\lfloor \left( 1+c\right)
k^{\ast }\right\rfloor \leq T_{m}$ for some $c>0$; we have, using the mean
value theorem
\begin{align*}
& r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \Delta _{m}^{\prime }
\sum_{t=m+k^{\ast }+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right\vert }{
m^{1/2}\left( 1+\frac{\widetilde{k}}{m}\right)
\left( \frac{\widetilde{k}}{m+\widetilde{k}}\right) ^{{\Greekmath 0111} }} \\
=& c_{0}r_{m}^{{\Greekmath 0111} -1/2}\frac{k^{\ast }+O_{P}\left( \left( k^{\ast }\right)
^{1/2}\right) }{m^{1/2}\left( 1+\frac{k^{\ast }}{m}\right)
\left( \frac{k^{\ast }}{m+k^{\ast }}\right) ^{{\Greekmath 0111} }}\mathbf{c}
_{1}^{\prime }\Delta _{m} \\
=& c_{0}r_{m}^{{\Greekmath 0111} -1/2}m^{1/2}\left( \frac{k^{\ast }}{m+k^{\ast }}\right)
^{1-{\Greekmath 0111} }\mathbf{c}_{1}^{\prime }\Delta _{m},
\end{align*}
which again immediately yields the desired result as long as ((ref)
) holds.
proof[Proof of Theorem (ref)]
We will study the limiting behaviour of
\begin{equation*}
r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq a_{m}}\frac{\left\vert Q\left(
m;k\right) \right\vert }{g_{{\Greekmath 0111} }\left( m;k\right) }=r_{m}^{{\Greekmath 0111} -1/2}\frac{
\left\vert \sum_{t=m+1}^{m+a_{m}}\widehat{{\Greekmath 010F} }
_{t}\right\vert }{m^{1/2}\left( 1+\frac{a_{m}}{m}\right)
\left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }},
\end{equation*}
under the alternative hypothesis of ((ref)). By standard
algebra, it holds that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
\widehat{{\Greekmath 010F} }_{t}\right\vert }{m^{1/2}\left( 1+\frac{a_{m}
}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }} \\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime
}\left( \sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime
}\right) ^{-1}\left( \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t}\right) +\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}
_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}.
\end{align*}
We begin by noting that, following the passages above, it is not hard to see
that
\begin{equation}
\left\Vert \sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}\right\Vert =O_{P}\left(
d^{1/2}a_{m}\right) .
\end{equation}
Using Lemmas (ref), ((ref)) and ((ref)), it follows
that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\left( \left( \frac{1}{m}\sum_{t=1}^{m}
\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }\right) ^{-1}-\mathbf{C}^{-1}\right)
\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t}\right) \right\vert }{m^{1/2}\left( 1+\frac{a_{m}}{m}
\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }} \\
\leq &r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\Vert \sum_{t=m+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\right\Vert \left\Vert \left( \frac{1}{m
}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }\right) ^{-1}-\mathbf{C}
^{-1}\right\Vert _{F}\left\Vert \frac{1}{m}\sum_{t=1}^{m}
\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}
\\
=&O_{P}\left( 1\right) r_{m}^{{\Greekmath 0111} -1/2}m^{-1/2}\left( \frac{a_{m}}{m}
\right) ^{-{\Greekmath 0111} }d^{1/2}a_{m}\frac{d}{m^{1/2}}\left( \frac{d}{m}\right)
^{1/2}=O_{P}\left( d^{2}\frac{a_{m}^{1/2}}{m}\right) =o_{P}\left( 1\right) ,
\end{align*}
by having $d=O\left( m^{1/4}\right) $. Therefore
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime
}\left( \sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime
}\right) ^{-1}\left( \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t}\right) +\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}
_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}
\\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }
\mathbf{C}^{-1}\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}
_{t}{\Greekmath 010F} _{t}\right) +\sum_{t=m+k^{\ast }+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+
\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}
\right) ^{{\Greekmath 0111} }}+o_{P}\left( 1\right) .
\end{align*}
Similarly, we note that
\begin{equation*}
E\left\Vert \sum_{t=m+1}^{m+a_{m}}\left( \mathbf{x}_{t}-\mathbf{c}
_{1}\right) \right\Vert ^{2}=\sum_{j=2}^{d}E\left(
\sum_{t=m+1}^{m+a_{m}}\left( x_{j,t}-E\left( x_{j,0}\right) \right) \right)
^{2}\leq c_{0}da_{m},
\end{equation*}
by using the fact that $x_{j,t}-E\left( x_{j,0}\right) $ is a weakly
dependent process in the sense of Definition (ref), and
Proposition 4 in berkes2011split, whence $\left\Vert
\sum_{t=m+1}^{m+a_{m}}\left( \mathbf{x}_{t}-\mathbf{c}_{1}\right)
\right\Vert =O_{P}\left( d^{1/2}a_{m}^{1/2}\right) $. Therefore, it is easy
to see that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}\left(
\mathbf{x}_{t}-\mathbf{c}_{1}\right) ^{\prime }\mathbf{C}^{-1}
\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right)
\right\vert }{m^{1/2}\left( 1+\frac{a_{m}}{m}\right)
\left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }} \\
\leq &r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\Vert \sum_{t=m+1}^{m+a_{m}}
\left( \mathbf{x}_{t}-\mathbf{c}_{1}\right) \right\Vert \left\Vert \mathbf{C}
^{-1}\right\Vert _{F}\left\Vert \frac{1}{m}\sum_{t=1}^{m}
\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}
\\
=&O_{P}\left( r_{m}^{{\Greekmath 0111} -1/2}m^{-1/2}\left( \frac{a_{m}}{m}\right) ^{-{\Greekmath 0111}
}d^{1/2}a_{m}^{1/2}d^{1/2}\left( \frac{d}{m}\right) ^{1/2}\right)
=O_{P}\left( \frac{d^{3/2}}{m^{1/2}}\right) =o_{P}\left( 1\right) ,
\end{align*}
having used\ Lemmas (ref) and (ref), ((ref))
and $d=O\left( m^{1/4}\right) $. This entails that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }
\mathbf{C}^{-1}\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}
_{t}{\Greekmath 010F} _{t}\right) +\sum_{t=m+k^{\ast }+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+
\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}
\right) ^{{\Greekmath 0111} }} \\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-a_{m}\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{c}
_{1}^{\prime }\mathbf{C}^{-1}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right) +
\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m}\right\vert }{m^{1/2}\left( 1+\frac{a_{m}}{m}\right)
\left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}+o_{P}\left(
1\right) \\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\frac{a_{m}}{m}\left(
\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right) +\sum_{t=m+k^{\ast
}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left(
1+\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{
m+a_{m}}\right) ^{{\Greekmath 0111} }}+o_{P}\left( 1\right) ,
\end{align*}
on account of the fact that $\mathbf{c}_{1}^{\prime }\mathbf{C}^{-1}\mathbf{x
}_{t}=1$ for all $t$. Also note that
\begin{equation*}
r_{m}^{{\Greekmath 0111} -1/2}\frac{a_{m}}{m}\frac{\left\vert \left( \frac{1
}{m^{1/2}}\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right) \right\vert }{\left( 1+
\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}
\right) ^{{\Greekmath 0111} }}=O_{P}\left( 1\right) r_{m}^{{\Greekmath 0111} -1/2}\left( \frac{a_{m}}{m
}\right) ^{1-{\Greekmath 0111} }=O_{P}\left( \left( \frac{a_{m}}{m}\right) ^{1/2}\right)
=o_{P}\left( 1\right) ,
\end{equation*}
whence
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-\frac{a_{m}}{m}\left(
\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right) +\sum_{t=m+k^{\ast
}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left(
1+\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{
m+a_{m}}\right) ^{{\Greekmath 0111} }} \\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}+\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}
_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}
+o_{P}\left( 1\right) .
\end{align*}
Finally, using Lemma (ref), it holds that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}-{\Greekmath 011B} W_{1,m}\left( a_{m}\right) \right\vert }{m^{1/2}\left(
1+\frac{a_{m}}{m}\right) \left( \frac{a_{m}}{
m+a_{m}}\right) ^{{\Greekmath 0111} }} \\
=&O_{P}\left( 1\right) r_{m}^{{\Greekmath 0111} -1/2}m^{{\Greekmath 0110} _{1}-1/2}\left( \frac{a_{m}
}{m}\right) ^{-{\Greekmath 0111} }=o_{P}\left( 1\right) ,
\end{align*}
so that
\begin{align*}
&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert \sum_{t=m+1}^{m+a_{m}}
{\Greekmath 010F} _{t}+\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}
_{t}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
a_{m}}{m}\right) \left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}
\\
=&r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert {\Greekmath 011B} W_{1,m}\left( a_{m}\right) +
\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m}\right\vert }{m^{1/2}\left( 1+\frac{a_{m}}{m}\right)
\left( \frac{a_{m}}{m+a_{m}}\right) ^{{\Greekmath 0111} }}+o_{P}\left(
1\right) \\
=&a_{m}^{-1/2}\left\vert {\Greekmath 011B} W_{1,m}\left( a_{m}\right) +\sum_{t=k^{\ast
}+1}^{a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert +o_{P}\left(
1\right) ,
\end{align*}
where the last result follows from having $a_{m}=o\left( m\right) $. Given
that the distribution of $W_{1,m}\left( \cdot \right) $ does not depend on $
m $, it follows that
\begin{equation*}
a_{m}^{-1/2}\left\vert {\Greekmath 011B} W_{1,m}\left( a_{m}\right) +\sum_{t=m+k^{\ast
}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert \overset{
\mathcal{D}}{=}\left\vert {\Greekmath 011B} W\left( 1\right)
+a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m}\right\vert ,
\end{equation*}
where $W\left( \cdot \right) $ is a standard Wiener. The results above
entail that, as $m\rightarrow \infty $
\begin{equation*}
P\left( r_{m}^{{\Greekmath 0111} -1/2}\max_{a_{m}\leq k\leq a_{m}}\frac{\left\vert
Q\left( m;a_{m}\right) \right\vert }{g\left( m;a_{m}\right) }\leq x\right)
=P\left( \left\vert {\Greekmath 011B} Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\Delta _{m}\right\vert \leq x\right) +o_{P}\left(
1\right) ,
\end{equation*}
for all $-\infty <x<\infty $, with $Z\sim N\left( 0,1\right) $.
We are now ready to prove the main statement of the theorem. By assumption, $
\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert =c_{0}\left\Vert
\mathbf{c}_{1}\right\Vert \left\Vert \Delta _{m}\right\Vert $, where $
c_{0}>0 $ is the cosine of the angle between $\mathbf{c}_{1}$ and $\Delta
_{m}$. Hence, Assumption (ref)(ii) entails
\begin{equation*}
\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert =c_{0}\left\Vert
\mathbf{c}_{1}\right\Vert \left\Vert \Delta _{m}\right\Vert =\Omega \left(
d^{1/2}\Delta _{m}\right) ;
\end{equation*}
henceforth, we can and will assume, without loss of generality, that $
\mathbf{c}_{1}^{\prime }\Delta _{m}>0$. We note that, using the fact that,
by Assumption (ref)(i), $a_{m}-k^{\ast }=\Omega
\left( a_{m}\right) $, we have
\begin{align}
&a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m} \\
=&a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{c}_{1}^{\prime }\Delta
_{m}+a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\left( \mathbf{x}_{t}-
\mathbf{c}_{1}\right) ^{\prime }\Delta _{m} \notag \\
=&a_{m}^{1/2}\Omega \left( d^{1/2}\Delta _{m}\right) +O_{P}\left(
d^{1/2}\Delta _{m}\right) , \notag
\end{align}
where the second term follows from the CLT, viz.
\begin{equation*}
\left\vert a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\left( \mathbf{x}_{t}-
\mathbf{c}_{1}\right) ^{\prime }\Delta _{m}\right\vert \leq \left\Vert
a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\left( \mathbf{x}_{t}-\mathbf{c}
_{1}\right) \right\Vert \left\Vert \Delta _{m}\right\Vert =O_{P}\left(
d^{1/2}\right) \Omega \left( \Delta _{m}\right) .
\end{equation*}
Equation ((ref)), and the fact that we are assuming $\mathbf{c}
_{1}^{\prime }\Delta _{m}>0$, imply that
\begin{equation*}
\lim_{m\rightarrow \infty }P\left( {\Greekmath 011B} Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast
}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}>0\right) =1,
\end{equation*}
and
\begin{equation}
a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m}\overset{\mathcal{P}}{\rightarrow }\infty ,
\end{equation}
as $m\rightarrow \infty $. Therefore, as $m\rightarrow \infty $
\begin{equation*}
P\left( r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert Q\left( m;a_{m}\right) \right\vert
}{g_{{\Greekmath 0111} }\left( m;a_{m}\right) }\leq x\right) =P\left( {\Greekmath 011B}
Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta
_{m}\leq x\right) +o_{P}\left( 1\right) .
\end{equation*}
Given that, by ((ref))
\begin{equation*}
\lim_{m\rightarrow \infty }P\left( {\Greekmath 011B} Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast
}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime }\Delta _{m}\leq c_{{\Greekmath 010B} ,{\Greekmath 0111}
}\right) =\lim_{m\rightarrow \infty }P\left( Z\leq \frac{c_{{\Greekmath 010B} ,{\Greekmath 0111} }}{
{\Greekmath 011B} }-\frac{1}{{\Greekmath 011B} }a_{m}^{-1/2}\sum_{t=m+k^{\ast }+1}^{m+a_{m}}
\mathbf{x}_{t}^{\prime }\Delta _{m}\right) =0,
\end{equation*}
putting all the above together, we finally have
\begin{equation*}
\lim_{m\rightarrow \infty }P\left( r_{m}^{{\Greekmath 0111} -1/2}\frac{\left\vert Q\left(
m;a_{m}\right) \right\vert }{g_{{\Greekmath 0111} }\left( m;a_{m}\right) }\leq c_{{\Greekmath 010B}
,{\Greekmath 0111} }\right) =0.
\end{equation*}
We now conclude, by noting that the event $\left\{ {\Greekmath 011C} _{m}>a_{m}\right\} $
is equivalent to having
\begin{equation*}
\left\{ r_{m}^{{\Greekmath 0111} -1/2}\left\vert Q\left( m;a_{m}\right) \right\vert \leq
c_{{\Greekmath 010B} ,{\Greekmath 0111} }g_{{\Greekmath 0111} }\left( m;a_{m}\right) \right\} .
\end{equation*}
We have shown that $\lim_{m\rightarrow \infty }P\left( {\Greekmath 011C}
_{m}>a_{m}\right) =0$, whence $\lim_{m\rightarrow \infty }P\left( {\Greekmath 011C}
_{m}=a_{m}\right) =1$.
proof[Proof of Theorem (ref)]
We prove that Lemmas (ref)-(ref) hold under ((ref)). Thereafter, the proofs of Theorems (ref), (ref)
and (ref) and Lemma (ref) can be repeated verbatim.
Further, for simplicity, we report the proof for the case $p=1$, writing (
(ref)) as
\begin{equation}
y_{t}={\Greekmath 011A} y_{t-1}+\mathbf{z}_{t}^{\prime }{\Greekmath 010C} ^{Z}+{\Greekmath 010F} _{t}.
\end{equation}
Equation ((ref)) can be expressed in recursive form
\begin{equation}
y_{t}={\Greekmath 011A} ^{t}y_{0}+\sum_{j=0}^{t-1}{\Greekmath 011A} ^{j}\left( \mathbf{z}
_{t-j}^{\prime }{\Greekmath 010C} ^{Z}+{\Greekmath 010F} _{t-j}\right) ,
\end{equation}
where $y_{0}$ is an initial condition. Under Assumption (ref)
(ii), ((ref)) admits a unique stationary, non-anticipative solution
given by
\begin{equation}
\overline{y}_{t}=\sum_{j=0}^{\infty }{\Greekmath 011A} ^{j}\left( \mathbf{z}
_{t-j}^{\prime }{\Greekmath 010C} ^{Z}+{\Greekmath 010F} _{t-j}\right) =\sum_{j=0}^{\infty }{\Greekmath 011A}
^{j}u_{t-j},
\end{equation}
where $u_{t}=\mathbf{z}_{t}^{\prime }{\Greekmath 010C} ^{Z}+{\Greekmath 010F} _{t}$, for any
(stochastically bounded) initial value $y_{0}$. By Assumption (ref)
, $x_{j,t}=h_{j}^{x}\left( {\Greekmath 0111} _{j,t}^{x},...\right) $ for $2\leq j\leq d-p$
and ${\Greekmath 010F} _{t}=h^{{\Greekmath 010F} }\left( {\Greekmath 0111} _{t}^{{\Greekmath 010F} },...\right) $;
hence, it immediately follows that we can use the representation $
u_{t}=h^{u}\left( {\Greekmath 0111} _{t}^{u},...\right) $ and that $u_{t}$ satisfies
Assumption (ref). Define the ${\Greekmath 011B} $-fields $\mathcal{F}_{t-\ell
}^{t-j}=\left\{ {\Greekmath 0111} _{i}^{u}\right\} _{i=t-\ell }^{t-j}$, with the
convention that $\mathcal{F}_{t-\ell }^{t-j}$ is empty if $j>\ell $, and $
\mathcal{F}_{t-\min \left\{ \ell ,j\right\} }^{-\infty }=\left\{ \widetilde{
{\Greekmath 0111} }_{t-\min \left\{ \ell ,j\right\} }^{u},...,\widetilde{{\Greekmath 0111} }_{-\infty
}^{u}\right\} $, where $\widetilde{{\Greekmath 0111} }_{t}^{u}$ is an independent copy of
${\Greekmath 0111} _{t}^{u}$, forming an i.i.d. sequence. We now define the
coupling constructions
\begin{equation*}
\widetilde{u}_{t-j,\ell }=h^{u}\left( \mathcal{F}_{t-\ell }^{t-j},\mathcal{F}
_{t-\min \left\{ \ell ,j\right\} }^{-\infty }\right) ,
\end{equation*}
and
\begin{equation*}
\widetilde{y}_{t,\ell }=\sum_{j=0}^{\infty }{\Greekmath 011A} ^{j}\widetilde{u}_{t-j,\ell
}.
\end{equation*}
Using Minkowski's inequality, we have that
\begin{equation*}
\left\vert y_{t}-\widetilde{y}_{t,\ell }\right\vert _{4}\leq \left\vert
y_{t}-\overline{y}_{t}\right\vert _{4}+\left\vert \overline{y}_{t}-
\widetilde{y}_{t,\ell }\right\vert _{4}=I+II,
\end{equation*}
with
\begin{equation}
\left\vert y_{t}-\overline{y}_{t}\right\vert _{4}\leq \left\vert {\Greekmath 011A}
^{t}y_{0}\right\vert _{4}+\left\vert \sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right\vert _{4}\leq c_{0}\left\vert {\Greekmath 011A} \right\vert
^{t}+\sum_{j=t}^{\infty }\left\vert {\Greekmath 011A} \right\vert ^{j}\left\vert
u_{0}\right\vert _{4}\leq c_{1}\left\vert {\Greekmath 011A} \right\vert ^{t},
\end{equation}
and
\begin{align}
& \left\vert \overline{y}_{t}-\widetilde{y}_{t,\ell }\right\vert _{4}
\\
\leq & \sum_{j=0}^{\infty }\left\vert {\Greekmath 011A} \right\vert ^{j}\left\vert
u_{t-j}-\widetilde{u}_{t-j,\ell }\right\vert _{4}=\sum_{j=0}^{\ell
}\left\vert {\Greekmath 011A} \right\vert ^{j}\left\vert u_{t-j}-\widetilde{u}_{t-j,\ell
}\right\vert _{4}+\sum_{j=\ell +1}^{\infty }\left\vert {\Greekmath 011A} \right\vert
^{j}\left\vert u_{t-j}-\widetilde{u}_{t-j,\ell }\right\vert _{4} \notag \\
\leq & \sum_{j=0}^{\ell }\left\vert {\Greekmath 011A} \right\vert ^{j}\left\vert u_{t-j}-
\widetilde{u}_{t-j,\ell }\right\vert _{4}+\left\vert {\Greekmath 011A} \right\vert ^{\ell
}\sum_{j=1}^{\infty }\left\vert {\Greekmath 011A} \right\vert ^{j}\left( \left\vert
u_{t-j}\right\vert _{4}+\left\vert \widetilde{u}_{t-j,\ell }\right\vert
_{4}\right) \notag \\
\leq & c_{0}\left( \sum_{j=0}^{\ell }\left\vert {\Greekmath 011A} \right\vert ^{j}\right)
\ell ^{-a}+c_{1}\left\vert {\Greekmath 011A} \right\vert ^{\ell }\leq c_{2}\ell ^{-a},
\notag
\end{align}
for some $c_{2}<\infty $. Note that equation ((ref)) entails that $
\overline{y}_{t}$ also satisfies Assumption (ref).
We are now ready to prove that Lemmas (ref)-(ref)
hold under ((ref)). Let $\overline{\mathbf{x}}_{t}=\left( \mathbf{z}
_{t}^{\prime },\overline{\mathbf{y}}_{p,t}^{\prime }\right) ^{\prime }$. We
begin with Lemma (ref), and note that
\begin{align}
& \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }-\frac{1}{m}
\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\overline{\mathbf{x}}_{t}^{\prime }
\\
=& \frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left( \mathbf{x}_{t}-
\overline{\mathbf{x}}_{t}\right) ^{\prime }+\frac{1}{m}\sum_{t=1}^{m}\left(
\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right) \overline{\mathbf{x}}
_{t}^{\prime }+\frac{1}{m}\sum_{t=1}^{m}\left( \mathbf{x}_{t}-\overline{
\mathbf{x}}_{t}\right) \left( \mathbf{x}_{t}-\overline{\mathbf{x}}
_{t}\right) ^{\prime }. \notag
\end{align}
It holds that
\begin{align*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left( \mathbf{x
}_{t}-\overline{\mathbf{x}}_{t}\right) ^{\prime }\right\Vert _{F}^{2}=&
\sum_{j,h=1}^{d}\left( \frac{1}{m}\sum_{t=1}^{m}\overline{x}_{j,t}\left(
x_{h,t}-\overline{x}_{h,t}\right) \right) ^{2} \\
\leq & \sum_{j=1}^{d}\left( \frac{1}{m}\sum_{t=1}^{m}\overline{x}
_{j,t}^{2}\right) \sum_{h=1}^{d}\left( \frac{1}{m}\sum_{t=1}^{m}\left(
x_{h,t}-\overline{x}_{h,t}\right) ^{2}\right) .
\end{align*}
It is easy to see that
\begin{equation*}
E\sum_{j=1}^{d}\left( \frac{1}{m}\sum_{t=1}^{m}\overline{x}_{j,t}^{2}\right)
\leq c_{0}d,
\end{equation*}
and
\begin{equation*}
E\sum_{h=1}^{d}\left( \frac{1}{m}\sum_{t=1}^{m}\left( x_{h,t}-\overline{x}
_{h,t}\right) ^{2}\right) \leq c_{0}dm^{-1},
\end{equation*}
because by ((ref)), for all $h$, $\left\vert x_{h,t}-\overline{x}
_{h,t}\right\vert _{2}\leq \left\vert x_{h,t}-\overline{x}_{h,t}\right\vert
_{4}\leq c_{1}\left\vert {\Greekmath 011A} \right\vert ^{t}$. Hence
\begin{equation*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left( \mathbf{x
}_{t}-\overline{\mathbf{x}}_{t}\right) ^{\prime }\right\Vert
_{F}=O_{P}\left( \frac{d}{m^{1/2}}\right) ,
\end{equation*}
and the same can be shown for the other terms in ((ref)); we note that
these bounds are not the sharpest possible (in essence, this is due to the
fact that some coordinates of $\mathbf{x}_{t}$ and $\overline{\mathbf{x}}
_{t} $ may be the same, thus reducing the dimensionality), but they suffice
for our purposes. Lemma (ref) now follows readily from repeating
the original proof, using $\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\overline{
\mathbf{x}}_{t}^{\prime }$ instead of $\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}
_{t}^{\prime }$. We now turn to Lemma (ref). We estimate
\begin{equation*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\left( \mathbf{x}_{t}-\overline{\mathbf{x
}}_{t}\right) {\Greekmath 010F} _{t}\right\Vert =\sum_{h=1}^{d}\left\vert
\sum_{t=1}^{m}\left( x_{h,t}-\overline{x}_{h,t}\right) {\Greekmath 010F}
_{t}\right\vert ^{2}\leq \sum_{h=1}^{d}\left\vert \sum_{t=1}^{m}{\Greekmath 011A}
^{t}y_{0}{\Greekmath 010F} _{t}\right\vert ^{2}+\sum_{h=1}^{d}\left\vert
\sum_{t=1}^{m}\left( {\Greekmath 010F} _{t}\sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right) \right\vert ^{2},
\end{equation*}
having used ((ref)) and ((ref)). Considering the first
term, it holds that
\begin{equation*}
E\sum_{h=1}^{d}\left\vert \sum_{t=1}^{m}{\Greekmath 011A} ^{t}y_{0}{\Greekmath 010F}
_{t}\right\vert ^{2}=\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left\vert {\Greekmath 011A}
\right\vert ^{t+s}E\left( y_{0}^{2}{\Greekmath 010F} _{t}{\Greekmath 010F} _{s}\right) \leq
\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left\vert {\Greekmath 011A} \right\vert ^{t+s}\left\vert
y_{0}\right\vert _{4}^{2}\left\vert {\Greekmath 010F} _{t}\right\vert _{4}\left\vert
{\Greekmath 010F} _{s}\right\vert _{4}\leq c_{0}d,
\end{equation*}
having used the Cauchy-Schwartz inequality (twice), Assumption (ref)
, and Assumption (ref). As far as the second term is concerned, note
that
\begin{align*}
& E\sum_{h=1}^{d}\left\vert \sum_{t=1}^{m}\left( {\Greekmath 010F}
_{t}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right) \right\vert ^{2} \\
=& E\sum_{h=1}^{d}\left\vert \sum_{t=1}^{m}\left( {\Greekmath 011A} ^{t}{\Greekmath 010F}
_{t}\sum_{h=0}^{\infty }{\Greekmath 011A} ^{h}u_{h}\right) \right\vert
^{2}=E\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left( {\Greekmath 011A} ^{t+s}{\Greekmath 010F} _{t}{\Greekmath 010F}
_{s}\sum_{h,i=0}^{\infty }{\Greekmath 011A} ^{h+i}u_{h}u_{i}\right) \\
=& \sum_{h=1}^{d}\sum_{t,s=1}^{m}{\Greekmath 011A} ^{t+s}\sum_{h,i=0}^{\infty }{\Greekmath 011A}
^{h+i}E\left( {\Greekmath 010F} _{t}{\Greekmath 010F} _{s}u_{h}u_{i}\right) \leq c_{0}d,
\end{align*}
which follows again from Assumptions (ref) and (ref)
(ii), and H\"{o}lder's inequality. Hence it follows that
\begin{equation*}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\left( \mathbf{x}_{t}-\overline{\mathbf{x
}}_{t}\right) {\Greekmath 010F} _{t}\right\Vert =O_{P}\left( \frac{d^{1/2}}{m}\right)
,
\end{equation*}
and now the desired result follows by repeating the proof of Lemma (ref) using $\overline{\mathbf{x}}_{t}$ instead of $\mathbf{x}_{t}$.
Finally, we consider Lemma (ref), and we estimate
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t})\right\Vert \leq
c_{0}d^{1/2}\left( \max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}
\left\vert \sum_{t=m+1}^{m+k}{\Greekmath 011A} ^{t}y_{0}\right\vert +\max_{1\leq k\leq
T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert \right) ,
\end{equation*}
where again we do not derive the sharpest bounds because we do not take into
account the fact that some elements of $\mathbf{x}_{t}$ and $\overline{
\mathbf{x}}_{t}$ coincide. It is easy to see that
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\vert
\sum_{t=m+1}^{m+k}{\Greekmath 011A} ^{t}y_{0}\right\vert =\left\vert y_{0}\right\vert
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\vert
\sum_{t=m+1}^{m+k}{\Greekmath 011A} ^{t}\right\vert =O_{P}\left( 1\right) ,
\end{equation*}
for all ${\Greekmath 0110} _{3}>0$. Also
\begin{align*}
& P\left( \max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert \geq
x\right) \\
\leq & P\left( \max_{0\leq \ell \leq \left\lceil \ln T_{m}\right\rceil
}\max_{\exp \left( \ell \right) \leq k\leq \exp \left( \ell +1\right) }\frac{
1}{k^{{\Greekmath 0110} _{3}}}\left\vert \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right\vert \geq x\right) \\
\leq & \sum_{\ell =0}^{\left\lceil \ln T_{m}\right\rceil }P\left( \max_{\exp
\left( \ell \right) \leq k\leq \exp \left( \ell +1\right) }\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert \geq x\exp
\left( {\Greekmath 0110} _{3}\ell \right) \right) \\
\leq & c_{0}x^{-p}\exp \left( -p{\Greekmath 0110} _{3}\ell \right) E\max_{\exp \left(
\ell \right) \leq k\leq \exp \left( \ell +1\right) }\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert ^{p} \\
\leq & x^{-p}\exp \left( -p{\Greekmath 0110} _{3}\ell \right) E\max_{1\leq k\leq \exp
\left( \ell +1\right) }\left\vert \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right\vert ^{p}
\end{align*}
for all $p\leq 4$. Note now that, by repeated application of Minkowski
inequality and by Assumptions (ref) and (ref)(ii)
\begin{align*}
& \left\vert \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right\vert _{p} \\
\leq & \sum_{t=m+1}^{m+k}\left\vert \sum_{j=t}^{\infty }{\Greekmath 011A}
^{j}u_{t-j}\right\vert _{p}\leq \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty
}\left\vert {\Greekmath 011A} \right\vert ^{j}\left\vert u_{t-j}\right\vert _{p} \\
\leq & \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }\left\vert {\Greekmath 011A} \right\vert
^{j}\left\vert u_{t-j}\right\vert _{p}\leq c_{0}\sum_{t=m+1}^{m+k}\left\vert
{\Greekmath 011A} \right\vert ^{t},
\end{align*}
so that
\begin{equation*}
\left\vert \sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert
_{p}^{p}\leq c_{0}\left( \sum_{t=m+1}^{m+k}\left\vert {\Greekmath 011A} \right\vert
^{t}\right) ^{p};
\end{equation*}
this is true for all $m$ and $k$, and by construction $\left\vert {\Greekmath 011A}
\right\vert ^{t}\geq 0$. Hence, we can apply Theorem F in
moricz1976moment, whence
\begin{equation*}
E\max_{1\leq k\leq \exp \left( \ell +1\right) }\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert ^{p}\leq
\log _{2}\left( 4\exp \left( \ell +1\right) \right) \left(
\sum_{t=m+1}^{m+k}\left\vert {\Greekmath 011A} \right\vert ^{t}\right) ^{p}\leq c_{0}\ell
,
\end{equation*}
which entails that
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\vert
\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty }{\Greekmath 011A} ^{j}u_{t-j}\right\vert
=O_{P}\left( 1\right) ,
\end{equation*}
for all ${\Greekmath 0110} _{3}>0$. Putting all together, for all ${\Greekmath 0110} _{3}>0$ it
holds that
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{1}{k^{{\Greekmath 0110} _{3}}}\left\Vert
\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t})\right\Vert
=O_{P}\left( d^{1/2}\right) ;
\end{equation*}
again, Lemma (ref) now follows from repeating the original proof
with $\overline{\mathbf{x}}_{t}$ instead of $\mathbf{x}_{t}$.
proof[Proof of Proposition (ref)]
Let ${\Greekmath 011B} =1$ for simplicity. We begin by showing that, on a suitably
enlarged probability space, there exist two independent standard Wiener
processes $\left\{ W_{1,m}\left( k\right) ,k\geq 1\right\} $ and $\left\{
W_{2,m}\left( k\right) ,k\geq 1\right\} $ such that
\begin{equation}
\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
\widehat{{\Greekmath 010F} }_{t}-\left( W_{1,m}\left( k\right) -\frac{k}{
m}W_{2,m}\left( m\right) \right) \right\vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}
_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right)
^{{\Greekmath 0111} _{j}}}=o_{P}\left( 1\right) .
\end{equation}
Some passages are repetitive, but we report them anyway to make the proof
easier to follow. We begin by noting that
\begin{align}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
\mathbf{x}_{t}^{\prime }\left( \left( \frac{1}{m}\sum_{t=1}^{m}
\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }\right) ^{-1}-\mathbf{C}^{-1}\right)
\left( \sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right)
\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right) \min_{1\leq
j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
\leq &\max_{1\leq k\leq T_{m}}\frac{\left\Vert
\sum_{t=m+1}^{m+k}\mathbf{x}_{t}\right\Vert \left\Vert \left(
\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime
}\right) ^{-1}-\mathbf{C}^{-1}\right) \right\Vert _{F}
\left\Vert \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert }{
m^{1/2}\left( 1+\frac{k}{m}\right) \min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \notag \\
=&O_{P}\left( 1\right) \max_{1\leq k\leq T_{m}}\frac{
d^{1/2}kdm^{-1/2}d^{1/2}m^{-1/2}}{m^{1/2}\left( 1+\frac{k}{m}
\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}
\notag \\
=&O_{P}\left( 1\right) \frac{d^{2}}{m^{1/2}}\max_{1\leq j\leq J}\frac{1}{
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}}\max_{1\leq k\leq
T_{m}}\left( \frac{k}{m+k}\right) ^{1-{\Greekmath 0111} _{j}}\frac{1}{d_{a_{m},{\Greekmath 0111} }}
\notag \\
=&O_{P}\left( 1\right) \frac{d^{2}}{m^{1/2}}=o_{P}\left( 1\right) , \notag
\end{align}
having used: Lemmas (ref) and (ref) in the third
line, and the fact that $d=o\left( m^{1/4}\right) $ in the last line.
Similarly, it holds that
\begin{align}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
\left( \mathbf{x}_{t}-\mathbf{c}_{1}\right) ^{\prime }\mathbf{C}^{-1}
\left( \frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F}
_{t}\right) \right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}
\\
\leq &\max_{1\leq k\leq T_{m}}\frac{\left\Vert
\sum_{t=m+1}^{m+k}\left( \mathbf{x}_{t}-\mathbf{c}_{1}\right) \right\Vert
\left\Vert \mathbf{C}^{-1}\right\Vert _{F}\left\Vert \frac{1}{m}
\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right\Vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{
m+k}\right) ^{{\Greekmath 0111} _{j}}} \notag \\
=&O_{P}\left( 1\right) \max_{1\leq k\leq T_{m}}\frac{k^{{\Greekmath 0110}
_{3}}d^{1/2}d^{1/2}d^{1/2}m^{-1/2}}{m^{1/2}\left( 1+\frac{k}{m}
\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}
\notag \\
=&O_{P}\left( 1\right) d^{3/2}\max_{1\leq j\leq J}\frac{1}{\widetilde{r}
_{m}^{1/2-{\Greekmath 0111} _{j}}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}}\max_{1\leq k\leq T_{m}}\left(
\frac{k}{m+k}\right) ^{{\Greekmath 0110} _{3}-{\Greekmath 0111} _{j}}\left( \frac{1}{m+k}\right)
^{1-{\Greekmath 0110} _{3}}\frac{1}{d_{a_{m},{\Greekmath 0111} }} \notag \\
=&O_{P}\left( 1\right) d^{3/2}\left( \frac{1}{m}\right) ^{1-{\Greekmath 0110} _{3}}
\widetilde{r}_{m}^{{\Greekmath 0110} _{3}-1/2}=o_{P}\left( 1\right) , \notag
\end{align}
having used: Lemma (ref) with ${\Greekmath 0110} _{3}=1/2+{\Greekmath 0122} $ for
all ${\Greekmath 0111} _{j}<1/2$\ and $1/2<{\Greekmath 0110} _{3}<\min \left\{ 1,{\Greekmath 0111} _{j}\right\} $
for all ${\Greekmath 0111} _{j}>1/2$, Lemma (ref) and ((ref)) in the
third line, and the fact that $d=o\left( m^{1/4}\right) $ in the last line.
Combining ((ref)) and ((ref)), and recalling that $\mathbf{c}
_{1}^{\prime }\mathbf{C}^{-1}\mathbf{x}_{t}=1$, we get
\begin{equation*}
\max_{1\leq k\leq T_{m}}\frac{\left\vert \left(
\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime }\right) \left(
\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime }\right) ^{-1}\left(
\sum_{t=1}^{m}\mathbf{x}_{t}{\Greekmath 010F} _{t}\right) -
\frac{k}{m}\sum_{t=1}^{m}{\Greekmath 010F} _{t}\right\vert }{
m^{1/2}\left( 1+\frac{k}{m}\right) \min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}=o_{P}\left( 1\right) ,
\end{equation*}
whence
\begin{align}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
\widehat{{\Greekmath 010F} }_{t}\right\vert }{m^{1/2}\left( 1+\frac{k}{m}
\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}
\\
=&\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
{\Greekmath 010F} _{t}-\frac{k}{m}\sum_{t=1}^{m}{\Greekmath 010F}
_{t}\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right)
\min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}}
+o_{P}\left( 1\right) . \notag
\end{align}
Also, by standard algebra based on Lemma (ref)
\begin{align*}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert \sum_{t=m+1}^{m+k}
{\Greekmath 010F} _{t}-W_{1,m}\left( k\right) \right\vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{
m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
=&O_{P}\left( 1\right) \max_{1\leq j\leq J}\max_{1\leq k\leq T_{m}}\frac{
k^{{\Greekmath 0110} _{1}}}{m^{1/2}\left( 1+\frac{k}{m}\right) \widetilde{r}
_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right)
^{{\Greekmath 0111} _{j}}} \\
=&O_{P}\left( 1\right) \max_{1\leq j\leq J}\frac{1}{\widetilde{r}
_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }}\max_{1\leq k\leq T_{m}}\frac{k^{{\Greekmath 0110}
_{1}-{\Greekmath 0111} _{j}}}{\left( m+k\right) ^{1/2-{\Greekmath 0111} _{j}}} \\
=&O_{P}\left( 1\right) a_{m}^{{\Greekmath 0110} _{1}-1/2}=o_{P}\left( 1\right) ,
\end{align*}
and
\begin{align*}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert \frac{k}{m}
\sum_{t=1}^{m}{\Greekmath 010F} _{t}-\frac{k}{m}
W_{2,m}\left( m\right) \right\vert }{m^{1/2}\left( 1+\frac{k}{m}
\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
=&O_{P}\left( 1\right) \max_{1\leq k\leq T_{m}}\frac{\frac{k}{m}m^{{\Greekmath 0110}
_{2}}}{m^{1/2}\left( 1+\frac{k}{m}\right) \min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
=&m^{{\Greekmath 0110} _{2}-1/2}O_{P}\left( 1\right) \max_{1\leq j\leq J}\frac{1}{
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }}\max_{1\leq k\leq
T_{m}}\left( \frac{k}{m+k}\right) ^{1-{\Greekmath 0111} _{j}}=m^{{\Greekmath 0110}
_{2}-1/2}O_{P}\left( 1\right) =o_{P}\left( 1\right) .
\end{align*}
Thence ((ref)) follows. We now study
\begin{align*}
&\max_{1\leq k\leq T_{m}}\frac{\left\vert W_{1,m}\left( k\right) -
\frac{k}{m}W_{2,m}\left( m\right) \right\vert }{m^{1/2}\left( 1+
\frac{k}{m}\right) \min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{k}{
m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
\overset{\mathcal{D}}{=}&\max_{1\leq k\leq T_{m}}\frac{\left\vert
W_{1}\left( k\right) -\frac{k}{m}W_{2}\left( m\right)
\right\vert }{m^{1/2}\left( 1+\frac{k}{m}\right) \min_{1\leq
j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
\overset{\mathcal{D}}{=}&\max_{1\leq k\leq T_{m}}\frac{\left( 1+
\frac{k}{m}\right) \left\vert W\left( \frac{k/m}{1+k/m}\right)
\right\vert }{\left( 1+\frac{k}{m}\right) \min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
\overset{\mathcal{D}}{=}&\max_{1\leq k\leq T_{m}}\frac{\left\vert W
\left( \frac{k/m}{1+k/m}\right) \right\vert }{\min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{k}{m+k}\right) ^{{\Greekmath 0111} _{j}}} \\
\overset{\mathcal{D}}{=}&\max_{1/m\leq t\leq T_{m}/m}\frac{\left\vert W
\left( \frac{t}{1+t}\right) \right\vert }{\min_{1\leq j\leq
J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }
\left( \frac{t}{1+t}\right) ^{{\Greekmath 0111} _{j}}},
\end{align*}
seeing as the distributions of $W_{1,m}\left( k\right) $ and $W_{2,m}\left(
k\right) $ do not depend on $m$ (second line), and by the same token as in (
(ref)), with $W\left( \cdot \right) $ standard Wiener. Therefore,
letting $J^{\ast }$ be the set of indices $j$ for which ${\Greekmath 0111} _{j}>1/2$ and
repeatedly using the scale transform of Wiener process
\begin{align*}
&\max_{1/m\leq t\leq T_{m}/m}\frac{\left\vert W\left( \frac{t}{
1+t}\right) \right\vert }{\min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}d_{a_{m},{\Greekmath 0111} }\left( \frac{t}{
1+t}\right) ^{{\Greekmath 0111} _{j}}} \\
\overset{\mathcal{D}}{=}&\max \left\{ \max_{j\notin J^{\ast }}\max_{1/m\leq
t\leq T_{m}/m}\frac{\left\vert W\left( \frac{t}{1+t}\right)
\right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\left( \frac{t}{1+t}\right)
^{{\Greekmath 0111} _{j}}},\max_{j\in J^{\ast }}r_{m}^{{\Greekmath 0111} _{j}-1/2}\max_{a_{m}/m\leq
t\leq T_{m}/m}\frac{\left\vert W\left( \frac{t}{1+t}\right)
\right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}\left( \frac{t}{1+t}\right)
^{{\Greekmath 0111} _{j}}}\right\} \\
\overset{\mathcal{D}}{=}&\max \left\{ \max_{j\notin J^{\ast }}\max_{1/\left(
m+1\right) \leq u\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left(
u\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{{\Greekmath 0111} _{j}}},\max_{j\in
J^{\ast }}r_{m}^{{\Greekmath 0111} _{j}-1/2}\max_{a_{m}/\left( m+a_{m}\right) \leq u\leq
T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left( u\right) \right\vert }{
c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{{\Greekmath 0111} _{j}}}\right\} \\
\overset{\mathcal{D}}{=}&\max \left\{ \max_{j\notin J^{\ast }}\max_{1/\left(
m+1\right) \leq u\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left(
u\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{{\Greekmath 0111} _{j}}},\max_{j\in
J^{\ast }}\max_{1\leq s\leq \left( T_{m}\left( m+a_{m}\right) \right)
/\left( a_{m}\left( m+T_{m}\right) \right) }\frac{\left\vert W\left(
s\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}s^{{\Greekmath 0111} _{j}}}\right\} \\
\overset{\mathcal{D}}{=}&\max \left\{ \max_{j\notin J^{\ast }}\max_{1/\left(
m+1\right) \leq u\leq T_{m}/\left( m+T_{m}\right) }\frac{\left\vert W\left(
u\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{{\Greekmath 0111} _{j}}},\max_{j\in
J^{\ast }}\max_{\left( a_{m}\left( m+T_{m}\right) \right) /\left(
T_{m}\left( m+a_{m}\right) \right) \leq u\leq 1}\frac{\left\vert W\left(
u\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{1-{\Greekmath 0111} _{j}}}\right\} \\
\overset{a.s}{\rightarrow }&\max \left\{ \max_{j\notin J^{\ast }}\sup_{0<u<1}
\frac{\left\vert W\left( u\right) \right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{{\Greekmath 0111}
_{j}}},\max_{j\in J^{\ast }}\sup_{0<u<1}\frac{\left\vert W\left( u\right)
\right\vert }{c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{1-{\Greekmath 0111} _{j}}}\right\} \overset{
\mathcal{D}}{=}\sup_{0<u<1}\frac{\left\vert W\left( u\right) \right\vert }{
\min_{1\leq j\leq J}c_{{\Greekmath 010B} ,{\Greekmath 0111} _{j}}u^{\widetilde{{\Greekmath 0111} }_{j}}},
\end{align*}
whence the desired result.
proof[Proof of Proposition (ref)]
We know from the proof of Theorem (ref) that the presence of power
depends only on the non-centrality induced by the break, so - following the
logic in that proof - we need to study
\begin{equation}
\frac{\left( \widetilde{k}-k^{\ast }\right) \left\vert \mathbf{c}
_{1}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
\widetilde{k}}{m}\right) \min_{1\leq j\leq J}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}\left( \frac{\widetilde{k}}{m+\widetilde{k}}\right) ^{{\Greekmath 0111}
_{j}}},
\end{equation}
where $\widetilde{k}$ is specified later. Under condition (i), let $
\widetilde{k}=a_{m}+k^{\ast }$; note that whenever $j\in J^{\ast }$, ((ref)) is of the same order as
\begin{equation*}
\frac{a_{m}\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert }{
m^{1/2}\min_{j\in J^{\ast }}\left( \frac{a_{m}}{m}\right)
^{{\Greekmath 0111} _{j}}}=\frac{a_{m}^{1-\max_{j\in J^{\ast }}{\Greekmath 0111} _{j}}\left\vert
\mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert }{m^{1/2-\max_{j\in J^{\ast
}}{\Greekmath 0111} _{j}}},
\end{equation*}
and seeing as
\begin{equation*}
\lim_{m\rightarrow \infty }\frac{a_{m}^{1-\max_{j\in J^{\ast }}{\Greekmath 0111}
_{j}}\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert }{
m^{1/2-\max_{j\in J^{\ast }}{\Greekmath 0111} _{j}}}\times \frac{1}{a_{m}^{1/2}\left\vert
\mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert }=\infty ,
\end{equation*}
on account of ((ref)), it follows that under condition (i)
\begin{equation*}
\frac{\left( \widetilde{k}-k^{\ast }\right) \left\vert \mathbf{c}
_{1}^{\prime }\Delta _{m}\right\vert }{m^{1/2}\left( 1+\frac{
\widetilde{k}}{m}\right) \min_{1\leq j\leq J}\widetilde{r}_{m}^{1/2-{\Greekmath 0111}
_{j}}\left( \frac{\widetilde{k}}{m+\widetilde{k}}\right) ^{{\Greekmath 0111}
_{j}}}\geq c_{0}a_{m}^{1/2}\left\vert \mathbf{c}_{1}^{\prime }\Delta
_{m}\right\vert ,
\end{equation*}
for some $c_{0}>0$, whence ((ref)) yields the final result. Under
conditions (ii) and (iii), let $\widetilde{k}=\left\lfloor
\left( 1+c_{0}\right) k^{\ast }\right\rfloor $; by standard algebra, it is
easy to see that ((ref)) is proportional to
\begin{equation*}
\frac{k^{\ast }\left\vert \mathbf{c}_{1}^{\prime }\Delta _{m}\right\vert }{
m^{1/2}\left( 1+\frac{k^{\ast }}{m}\right) \min_{1\leq j\leq J}
\widetilde{r}_{m}^{1/2-{\Greekmath 0111} _{j}}\left( \frac{k^{\ast }}{
m+k^{\ast }}\right) ^{{\Greekmath 0111} _{j}}},
\end{equation*}
whence ((ref)) yields the desired result.