# Testing for high-dimensional white noise

Long Feng
School of Statistics and Data Science, Nankai University
and
Binghui Liu
School of Mathematics and Statistics & KLAS, Northeast Normal University
and
Yanyuan Ma
Department of Statistics, Pennsylvania State University

###### Abstract

Testing for multi-dimensional white noise is an important subject in
statistical inference. Such test in the high-dimensional case becomes an open
problem waiting to be solved, especially when the dimension of a time series
is comparable to or even greater than the sample size. To detect an arbitrary
form of departure from high-dimensional white noise, a few tests have been
developed. Some of these tests are based on max-type statistics, while others
are based on sum-type ones. Despite the progress, an urgent issue awaits to be
resolved: none of these tests is robust to the sparsity of the serial
correlation structure. Motivated by this, we propose a Fisher’s combination
test by combining the max-type and the sum-type statistics, based on the
established asymptotically independence between them. This combination test
can achieve robustness to the sparsity of the serial correlation structure,
and combine the advantages of the two types of tests. We demonstrate the
advantages of the proposed test over some existing tests through extensive
numerical results and an empirical analysis.

Keywords: Asymptotically independence, Fisher’s combination test, High-
dimensional white noise, Hypothesis test, Robustness

##  1 Introduction

Testing for white noise or serial correlation is an important problem in
statistical modeling and inference, especially in diagnostic checking for
linear regression and linear time series modeling. In recent years,
researchers are increasingly interested in modeling high-dimensional time
series data, which are becoming one of the most common data types, and
frequently appear in many applications, including meteorology, genomics,
chemometrics, biological and environmental research, finance and econometrics,
etc. This brings further challenge to diagnostic checking, as we need to
perform test for high-dimensional white noise, where the dimension of time
series is comparable to or even greater than the sample size, i.e. the
observed length of the time series.

For univariate time series, many widely used white noise tests have been
proposed in the literature (Li, 2004). Some of these tests have been extended
for testing multivariate time series (Hosking, 1980; Li and Mcleod, 1981),
which are, however, only suitable for the case that the dimension of the time
series is small compared to the sample size. Specifically, for univariate time
series, the celebrated Box-Pierce portmanteau test and its variations are
considered to be among the most popular omnibus tests for detecting non-
specific forms of deviation from white noise. These tests are particularly
convenient in practical applications, due to the fact that they are
asymptotically distribution-free and $\chi^{2}$-distributed under the null
hypothesis (Li, 2004; Lütkepohl, 2005). However, it is widely known that when
extended to the multivariate cases, these tests suffer the issue of slow
convergence to their asymptotic null distributions (Li et al., 2019).

Recently, multivariate white noise tests have undergone rapid development.
Some new omnibus tests, such as the tests proposed by Chang, Yao and Zhou
(2017), Li et al. (2019) and Tsay (2020) respectively, can even deal with
high-dimensional time series, where the dimension of the time series is
comparable to or even greater than the sample size. Specifically, Chang, Yao
and Zhou (2017) proposed a max-type test for high-dimensional white noise,
using the maximum absolute auto-correlations and cross-correlations of the
component series. Based on an approximation by the $L_{\infty}$-norm of a
normal random vector, the critical value of the max-type test can be evaluated
by bootstrapping from a multivariate normal distribution. Subsequently, Tsay
(2020) proposed a rank-based max-type test for high-dimensional white noise by
using Spearman’s rank correlation, and established the limiting null
distribution based on the theory of extreme values. On the other hand, Li et
al. (2019) proposed a sum-type test for high-dimensional white noise, using
the sum of the squared singular values of several lagged sample autocovariance
matrices. Using the random matrix theory, the asymptotic normality for the
test statistic under the null is established under the Marčenko-Pastur
asymptotic regime.

In general, the max-type test performs well in the case of sparse
correlations, i.e. there is a small amount of large absolute auto- or cross-
correlations at any nonzero lag. In contrast, the sum-type test performs well
in the case of non-sparse correlations, which encapsulates the serial
correlations within and across all component series. These two types of tests
have their own applicability, but neither of them can perform well in both
cases. In other words, neither of these two types of tests is applicable in
the case of sparse serial correlations. This motivates us to establish a new
test, which can combine the advantages of both types and is therefore
applicable to sparse and non-sparse serial correlations. To this end, we
reconsider the max-type test and the sum-type test, establish their asymptotic
independence, then combine them to construct a combination test. It should be
noted that the general idea of constructing combination tests by establishing
the asymptotic independence of max-type and sum-type statistics has appeared
in independence tests and covariance matrix tests for high-dimensional random
vectors, for example, in Li and Xue (2015) and Yu et al. (2020).

To combine the asymptotically independent tests, we employ the framework of
combining the p-values of independent tests (Littell and Folks, 1971). In a
great deal of early literatures, the problem of combining independent tests of
hypotheses has been widely considered, such as in Pearson (1938), Fisher
(1950), Wilk and Shapiro (1968) and Naik (1969), to name but a few. Among
these methods, the well known Fisher’s combination test proposed in Fisher
(1950) is usually regarded as one of the best choices, whose advantages were
discussed in Littell and Folks (1971). It should be noted that in addition to
the way of combining p-values of independent tests, there are other ways for
combining independent tests. For example, if all test statistics
asymptotically follow Gaussian distributions, a linear combination of the
statistics can be used to construct a combined test statistic. However, in
situation where the test statistics have different types of asymptotic
distributions, such as normal distribution and Gumble distribution, it is
usually difficult to directly combine these statistics, hence the way of
combining p-values becomes more practical.

In this paper, for testing high-dimensional white noise, we propose a Fisher’s
combination test by combining the p-values of the max-type and sum-type tests,
which is suitable to detect sparse and non-sparse serial correlations. Using
the extreme value theory and the martingale’s central limit theorem, we
establish the limiting null distributions of the max-type and sum-type
statistics, respectively. Then, we establish the asymptotically independence
between the two statistics under the null hypothesis, which enables us to use
Fisher’s framework of combining independent tests. Furthermore, we demonstrate
the advantages of the proposed Fisher’s combination test over its competitors
through extensive numerical results. In the empirical application, we
demonstrate the robust performance of the proposed Fisher’s combination test.

The main contributions of this paper are threefold as follows.

  * •

First, we established the limiting null distribution of the max-type statistic
for testing high-dimensional white noise, proved that this max-type test is
rate-optimal and investigated its local power function in some special cases.

  * •

Then, we proposed a new sum-type test for testing high-dimensional white
noise, where the relationship between the sample size and the dimension is not
constrained, while the existing sum-type test needs to impose the
Mar$\breve{\textrm{c}}$enko-Pastur regime, i.e. the ratio of the sample size
to the dimension goes to a constant.

  * •

Finally, we proved the asymptotic independence between the above max-type and
sum-type test statistics under both Gaussian and non-Gaussian distributions,
which is  the most important contribution of this paper. Based on this
asymptotic independence, we constructed the Fisher’s combination test that is
suitable to detect either sparse or non-sparse serial correlations. In
particular, we can eliminate the Gaussianity requirement of the error
distribution in establishing the asymptotic independence. In contrast, all the
existing literatures on establishing such asymptotic independence are limited
by the Gaussianity requirement. Indeed, the proof of such asymptotic
independence is more challenging than that of the asymptotic independence
between the max-type and sum-type test statistics proposed for many other
important high-dimensional testing problems, such as the high-dimensional
cross-sectional independence testing problem (Feng et al., 2022), the high-
dimensional location testing problem (Xu et al., 2016) and the high-
dimensional covariance matrix testing problem (Li and Xue, 2015; Yu et al.,
2020), which are limited by the Gaussianity requirement.

The rest of this paper is organized as follows. In Section 2, we describe the
problem of testing for high-dimensional white noise, reconsider the max-type
and sum-type tests, establish their asymptotic independence and then construct
the Fisher’s combination test. In Section 3, we present extensive numerical
results of the proposed test in comparison with some of its competitors,
followed by an empirical application in Section 4. Then, we conclude the paper
with some discussions in Section 5, and relegate the technical proofs to the
supplementary.

##  2 Methodology

###  2.1 Testing problem

Let $\\{\bm{\varepsilon}_{t}\\}$ be a $p$-dimensional weakly stationary time
series with mean zero, where
$\bm{\varepsilon}_{t}=(\varepsilon_{t1},\cdots,\varepsilon_{tp})^{\top}\in\mathbb{R}^{p}$.
Let $\bm{\Sigma}(k)=\left\\{{\sigma}_{ij}(k)\right\\}_{1\leqslant i,j\leqslant
p}\doteq\mathrm{cov}(\bm{\varepsilon}_{t+k},\bm{\varepsilon}_{t})$ denote the
autocovariance of $\bm{\varepsilon}_{t}$ at lag $k$, and let
$\bm{\Gamma}(k)=\left\\{{\rho}_{ij}(k)\right\\}_{1\leqslant i,j\leqslant
p}\doteq\mathrm{diag}\\{\bm{\Sigma}(0)\\}^{-1/2}\bm{\Sigma}(k)\mathrm{diag}\\{\bm{\Sigma}(0)\\}^{-1/2}$
denote the autocorrelation of $\bm{\varepsilon}_{t}$ at lag $k$, where for any
matrix ${\bf M}$, $\mathrm{diag}({\bf M})$ denotes the diagonal matrix
consisting of the diagonal elements of ${\bf M}$ only. Let
${\bm{\Sigma}}=\\{\sigma_{ij}\\}_{1\leqslant i,j\leqslant p}=\bm{\Sigma}(0)$
and $\bm{\Gamma}=\\{\rho_{ij}\\}_{1\leqslant i,j\leqslant p}=\bm{\Gamma}(0)$.
Let $\sigma_{i}^{2}=\sigma_{ii}$, for each $i\in\\{1,\cdots,p\\}$.
$\\{\bm{\varepsilon}_{t}\\}$ is white noise, if $\bm{\Sigma}(k)\equiv 0$ for
all $k\neq 0$.

With the observations
$\\{\bm{\varepsilon}_{1},\ldots,\bm{\varepsilon}_{n}\\}$, let

$\hat{\bm{\Gamma}}(k)=\left\\{\hat{\rho}_{ij}(k)\right\\}_{1\leqslant
i,j\leqslant
p}\doteq\operatorname{diag}\\{\hat{\bm{\Sigma}}(0)\\}^{-1/2}\hat{\bm{\Sigma}}(k)\operatorname{diag}\\{\hat{\bm{\Sigma}}(0)\\}^{-1/2}$

denote the sample autocorrelation matrix at lag $k$, where

$\hat{\bm{\Sigma}}(k)=\left\\{\hat{\sigma}_{ij}(k)\right\\}_{1\leqslant
i,j\leqslant
p}\doteq\frac{1}{n}\sum_{t=1}^{n-k}\bm{\varepsilon}_{t+k}\bm{\varepsilon}_{t}^{\mathrm{T}}$

denotes the sample autocovariance matrix at lag $k$.

We consider the following testing problem:

$\displaystyle H_{0}:\\{\bm{\varepsilon}_{t}\\}\mbox{~{}is white
noise}~{}~{}v.s.~{}~{}H_{1}:\\{\bm{\varepsilon}_{t}\\}\mbox{~{}is not white
noise,}$ (1)

where the dimension of time series $p$ is comparable to or even greater than
the sample size $n$.

###  2.2 The max-type test

Before proposing the Fisher’s combination test for testing high-dimensional
white noise, we need to re-examine the max-type and sum-type tests, which will
be combined to construct the combination test.

Since $\bm{\Gamma}(k)=0$ for any $k\geqslant 1$ under $H_{0}$, the max-type
test statistic $T_{\textrm{MAX}}$ is defined as

$\displaystyle T_{\textrm{MAX}}\doteq\max_{1\leqslant k\leqslant K}T_{n,k},$
(2)

which was first proposed by Chang, Yao and Zhou (2017), where
$T_{n,k}\doteq\max_{1\leqslant i,j\leqslant
p}n^{1/2}\left|\hat{\rho}_{ij}(k)\right|$ and $K\geqslant 1$ is an integer.
For this max-type test statistic, Chang, Yao and Zhou (2017) evaluated the
critical value by bootstrapping from a multivariate normal distribution, which
is a widely recognized practice in the case of sparse correlations.

To establish the Fisher’s combination test in this paper, we first derive the
limiting null distribution of the max-type test statistic, which will be
presented in the following Theorem 1. Specifically, Theorem 1 states that
$T_{\textrm{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})$ has an asymptotic
extreme-value distribution when both $n$ and $p$ go to infinity. Hence, a
level-$\alpha$ test with $\alpha\in(0,1)$ will be performed by rejecting
$H_{0}$ when $T_{\textrm{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})$ is larger
than $q_{\alpha}$, i.e. the $1-\alpha$ quantile of
$G(y)\doteq\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}$.

In deriving Theorem 1, we impose the following three conditions.

  * (C1)

$\varepsilon_{ti}$’s have one of the following two types of tails: (i) sub-
gaussian-type tails, i.e. there exist some constant $\eta>0$ and $M>0$, such
that $\mathbb{E}e^{\eta\varepsilon_{ti}^{2}/\sigma_{i}^{2}}\leq M$ for all
$i\in\\{1,\cdots,p\\}$ and $t\in\\{1,\cdots,n\\}$, where $p$ satisfies $\log
p=o(n^{1/5})$; (ii) polynomial-type tails, i.e. for some $\gamma_{0}$ and
$c_{1}>0$, $p\leq c_{1}n^{\gamma_{0}}$ and for some $\epsilon>0$ and $M>0$,
$\mathbb{E}|\varepsilon_{ti}/\sigma_{i}|^{4\gamma_{0}+4+\epsilon}\leq M$ for
all $i\in\\{1,\cdots,p\\}$ and $t\in\\{1,\cdots,n\\}$.

  * (C2)

There exists a positive constant $C$ such that $C^{-1}\leq\min_{1\leq i\leq
p}\sigma_{i}^{2}\leq\max_{1\leq i\leq p}\sigma_{i}^{2}\leq C$.

  * (C3)

There exists $\varrho\in(0,1)$ s.t. $|\rho_{ij}|\leq\varrho$ for all $1\leq
i<j\leq p$ with $p\geq 2$. $|C_{p}|/p^{2}\to 0$ as $p\to\infty$ if (C1)-(i)
holds; and $|C_{p}|/n^{\epsilon/8}\to 0$ if (C1)-(ii) holds. Here
$C_{p}\doteq\big{\\{}(i,j):|B_{p,(i,j)}|\geq p^{\kappa_{p}}\big{\\}}$ and
$B_{p,(i,j)}\doteq\big{\\{}(s,l):|\rho_{ij}\rho_{sl}|\geq\delta_{p}\big{\\}}$
for $1\leq i,j\leq p$ with $\delta_{p},\kappa_{p}>0$, $\delta_{p}=o(1/\log p)$
and $\kappa_{p}=o(1)$ as $p\to\infty$.

######  Remark 1.

Condition (C1) requires that the tail of the distributions of
$\varepsilon_{ti}$’s is sub-gaussian-type or polynomial-type, which is the
same as Condition (C2) or (C2∗) used in Cai et al. (2013). It is a more
general moment condition than the normal distribution assumption. Condition
(C2) requires that all the variances of $\varepsilon_{ti}$’s are bounded.
Condition (C3) requires that the number of variable pairs with strong
correlation cannot be too large.

######  Theorem 1.

Suppose Conditions (C1)-(C3) hold. Then, under $H_{0}$, for any
$y\in\mathbb{R}$, we have

$\displaystyle\operatorname{P}\left\\{T_{\textrm{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leqslant
y\right\\}\to G(y)$

as $n,p\to\infty$, where $G(y)=\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}$.

We recall that all technical proofs of the theorems and the proposition are
relegated to the supplementary.

Let $\mathcal{U}(c)$ be a set of matrices indexed by a constant $c$, which is
given by

$\left[\big{\\{}\bm{\Gamma}(1),\cdots,\bm{\Gamma}(K)\big{\\}}\in\mathbb{R}^{p\times
Kp}:\max_{1\leq k\leq K,1\leqslant i<j\leqslant
p}\left|\rho_{ij}(k)\right|\geqslant c(\log p/n)^{1/2}\right].$

Consider the following sparse alternative

$\displaystyle
H_{\mathrm{a}}^{R}(c)\doteq\left[F(\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}):\big{\\{}\mathrm{cor}_{F}(\bm{\varepsilon}_{t+1},\bm{\varepsilon}_{t}),\cdots,\mathrm{cor}_{F}(\bm{\varepsilon}_{t+K},\bm{\varepsilon}_{t})\big{\\}}\in\mathcal{U}(c)\right],$
(3)

where $F(\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n})$ denotes the joint
distribution of $\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}\\}$ and
$\mathrm{cor}_{F}$ denotes the autocorrelation matrix under the joint
distribution $F$. Let $\mathcal{T}_{\alpha}$ denote the set of all measurable
size-$\alpha$ tests.

Further, the following theorem characterizes the conditions under which the
power of the proposed max-type test
$\mathbb{I}\\{T_{\textrm{MAX}}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\\}$ tends to 1 as $n\to\infty$, under the alternative
$H_{\mathrm{a}}^{R}\left(b_{0}\right)$ for some constant $b_{0}$.

######  Theorem 2.

Assume that $\\{\bm{\varepsilon}_{t}\\}$ is a strictly stationary time series
and the long-run variance
$\gamma_{i}^{L}\doteq\lim_{n\to\infty}\mathrm{var}\left(n^{-1/2}\sum_{t=1}^{n}\varepsilon_{it}^{2}\right)$
is bounded for all $1\leq i\leq p$. Then, we have

$\inf_{F(\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n})\in
H_{\mathrm{a}}^{R}\left(b_{0}\right)}\operatorname{P}\left[\mathbb{I}\\{T_{\operatorname{MAX}}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\\}=1\right]=1-o(1),$

for all $b_{0}>3$, where the infimum is taken over the joint distribution
family $H_{\mathrm{a}}^{R}\left(b_{0}\right)$ of
$\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}\\}$ defined in (3).

Theorem 2 indicates that the above max-type test can detect alternatives of
order $(\log p/n)^{1/2}$. In Theorem 3, we will show that this test is rate-
optimal, i.e. the rate of the signal gap, $(\log p/n)^{1/2}$, cannot be
further relaxed.

######  Theorem 3.

Suppose $c_{0}<1$ is a positive constant, and let $\beta$ be a positive
constant satisfying $\alpha+\beta<1$. If $\log p/n=o(1)$, we have

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{F(\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n})\in
H_{\mathrm{a}}^{R}\left(c_{0}\right)}\operatorname{P}\left(T_{\alpha}=0\right)\geqslant
1-\alpha-\beta$

as $n,p\to\infty$, where the supremum is taken over the joint distribution
family $H_{\mathrm{a}}^{R}\left(c_{0}\right)$ of
$\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}\\}$ defined in (3).

Theorem 3 indicates that any measurable size-$\alpha$ test cannot
differentiate between the null hypothesis $H_{0}$ and the sparse alternative,
when

${\color[rgb]{0,0,0}\max_{1\leq k\leq K,1\leqslant i<j\leqslant
p}\left|\rho(k)_{ij}\right|<c_{0}(\log p/n)^{1/2}}$

for some constant $c_{0}<1$.

We now consider a special case to investigate the power function of the max-
type test when $c_{0}\in[1,3]$. Let $\varepsilon_{t1}=z_{t1}+\rho z_{t-1,1}$,
where $z_{t1}\sim\mathcal{N}(0,1)$ and $\rho=O(\sqrt{\log p/n})$. For
$i\in\\{2,\cdots,p\\}$, $\varepsilon_{ti}$’s are all i.i.d. from
$\mathcal{N}(0,1)$. $\\{\varepsilon_{t1}\\}_{t=1,\cdots,n}$ are independent of
$\\{\varepsilon_{ti}\\}_{t=1,\cdots,n},~{}i=2,\cdots,p$. By the Central Limit
Theorem, we have that
$\sqrt{n-k}\hat{\rho}_{ij}(k)\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1)$
for $k>1$;
$\sqrt{n-1}\hat{\rho}_{ij}(1)\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1)$
for $i\not=j$ and
${\color[rgb]{0,0,0}\frac{\sqrt{n-1}}{\sqrt{1-\frac{3\rho^{2}+2\rho^{4}+3\rho^{6}}{(1+\rho^{2})^{4}}}}\\{\hat{\rho}_{11}(1)-\frac{\rho}{1+\rho^{2}}\\}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1)}$
for $k=1$. Define $x_{\alpha}=2\log(Kp^{2})-\log\log(Kp^{2})+q_{\alpha}$.
Define $\mathcal{A}=\\{(i,j,k)|1\leq i,j\leq p,1\leq k\leq K\\}$. Thus,

$\displaystyle\textrm{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right\\}$ $\displaystyle=$
$\displaystyle\textrm{P}\left\\{\max_{1\leq i,j\leq p,1\leq k\leq
K}(n-k)\hat{\rho}^{2}_{ij}(k)\geq x_{\alpha}\right\\}$ $\displaystyle\geq$
$\displaystyle\textrm{P}\left\\{(n-1)\hat{\rho}^{2}_{11}(1)\geq
x_{\alpha}\right\\}$ $\displaystyle=$
$\displaystyle\textrm{P}\left\\{\left|\mathcal{N}\left(\frac{\sqrt{n}\rho}{{1+\rho^{2}}},1-{\color[rgb]{0,0,0}\frac{3\rho^{2}+2\rho^{4}+3\rho^{6}}{(1+\rho^{2})^{4}}}\right)\right|\geq\sqrt{x_{\alpha}}\right\\}+o(1)$
$\displaystyle=$
$\displaystyle\textrm{P}\left\\{\left|\mathcal{N}\left(\sqrt{n}\rho,1\right)\right|\geq\sqrt{x_{\alpha}}\right\\}+o(1)$
$\displaystyle=$
$\displaystyle\Phi(\sqrt{n}\rho-\sqrt{x_{\alpha}})+\Phi(-\sqrt{n}\rho-\sqrt{x_{\alpha}})+o(1)$

for sufficiently small $\rho$ and diverging $p$, where
$\mathcal{N}(\mu,\sigma^{2})$ denotes a random variable that follows the
normal distribution with mean $\mu$ and variance $\sigma^{2}$.

Additionally,

$\displaystyle\textrm{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right\\}$ $\displaystyle=$
$\displaystyle\textrm{P}\left\\{\max_{1\leq i,j\leq p,1\leq k\leq
K}(n-k)\hat{\rho}^{2}_{ij}(k)\geq x_{\alpha}\right\\}$ $\displaystyle\leq$
$\displaystyle\textrm{P}\left\\{(n-1)\hat{\rho}^{2}_{11}(1)\geq
x_{\alpha}\right\\}+\textrm{P}\left\\{\max_{(i,j,k)\in\mathcal{A}/{(1,1,1)}}(n-k)\hat{\rho}^{2}_{ij}(k)\geq
x_{\alpha}\right\\}$ $\displaystyle=$
$\displaystyle\Phi(\sqrt{n}\rho-\sqrt{x_{\alpha}})+\Phi(-\sqrt{n}\rho-\sqrt{x_{\alpha}})+\alpha+o(1).$

In the last equality, we used the following derivation. By the proof of
Theorem 1 in Chang et al. (2018), we have

$\displaystyle\textrm{P}\left\\{\max_{(i,j,k)\in\mathcal{A}/{(1,1,1)}}(n-k)\hat{\rho}^{2}_{ij}(k)\geq
x_{\alpha}\right\\}-\textrm{P}\left(\max_{1\leq l\leq Kp^{2}-1}\xi_{l}^{2}\geq
x_{\alpha}\right)\to 0,$

where $(\xi_{1},\cdots,\xi_{Kp^{2}-1})$ follows the multivariate normal
distribution with zero mean and the same correlation matrix of
$\\{(n-k)\hat{\rho}^{2}_{ij}(k)\\}_{(i,j,k)\in\mathcal{A}/{(1,1,1)}}$. After
some calculations, we have
$\mathrm{cor}\left\\{(n-k)\hat{\rho}_{11}(k),(n-k-1)\hat{\rho}_{11}(k+1)\right\\}\to
2\rho$,
$\mathrm{cor}\left\\{(n-k)\hat{\rho}_{11}(k),(n-k-2)\hat{\rho}_{11}(k+2)\right\\}\to\frac{\rho^{2}}{1+\rho^{2}}$
and the other correlations between $(n-k)\hat{\rho}^{2}_{ij}(k)$ are all
zeros. By Theorem 1 in the supplementary and condition $\rho=O(\sqrt{\log
p/n})$, we have $\textrm{P}\left(\max_{1\leq l\leq Kp^{2}-1}\xi_{l}^{2}\geq
x_{\alpha}\right)\to\alpha$.

Hence, the power function of the max-type test is

$\displaystyle\lim_{n,p\to\infty}\Phi(\sqrt{n}\rho-\sqrt{x_{\alpha}})$
$\displaystyle+\lim_{n,p\to\infty}\Phi(-\sqrt{n}\rho-\sqrt{x_{\alpha}})\leq\beta_{\textrm{MAX}}(\rho)$
$\displaystyle\leq$
$\displaystyle\alpha+\lim_{n,p\to\infty}\Phi(\sqrt{n}\rho-\sqrt{x_{\alpha}})+\lim_{n,p\to\infty}\Phi(-\sqrt{n}\rho-\sqrt{x_{\alpha}}).$

Note that $x_{\alpha}\sim 2\sqrt{\log p}$. If $\rho=c_{0}\sqrt{\log p/n}$, we
have: (1) if $0<c_{0}<2$, $c_{0}\sqrt{\log p}-\sqrt{x_{\alpha}}\to-\infty$,
$-c_{0}\sqrt{\log p}-\sqrt{x_{\alpha}}\to-\infty$ and
$\beta_{\textrm{MAX}}(\rho)\in(0,\alpha)$; (2) if $c_{0}>2$, $c_{0}\sqrt{\log
p}-\sqrt{x_{\alpha}}\to\infty$, $-c_{0}\sqrt{\log
p}-\sqrt{x_{\alpha}}\to-\infty$ and $\beta_{\textrm{MAX}}(\rho)=1$. On the
other hand, if $\sqrt{n}\rho=\sqrt{4\log p+c_{1}\sqrt{\log p}}$, we have
$\sqrt{n}\rho-\sqrt{x_{\alpha}}\to c_{1}/4$ and
$-\sqrt{n}\rho-\sqrt{x_{\alpha}}\to-\infty$, then
$\beta_{\textrm{MAX}}(\rho)\in\left(\Phi(c_{1}/4),\alpha+\Phi(c_{1}/4)\right)$.

######  Remark 2.

As mentioned in Chang, Zhou and Wang (2017), the max-type tests based on
asymptotic Gumble distributions usually have conservative size performance.
The proposed MAX test also has such limitation. To solve this problem, some
resampling methods can be employed, such as the bootstrap procedure used in
the max-type test proposed by Chang, Yao and Zhou (2017). The resampling
methods can also relax the conditions imposed on $\bm{\varepsilon}_{t}$’s
(Chang, Yao and Zhou, 2017). Unfortunately, such methods generally need to pay
a much heavier computational cost, especially in high-dimensional situations.
Hence, whether to use a test based on asymptotic distribution or based on
resampling strongly depends on the requirement of computational efficiency.

###  2.3 The sum-type test

In this subsection, we reconsider the sum-type test, with the test statistic
defined as

$\displaystyle
T_{\operatorname{SUM}}\doteq\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}\bm{\varepsilon}_{t}^{\top}\bm{\varepsilon}_{s}\bm{\varepsilon}_{t+l}^{\top}\bm{\varepsilon}_{s+l}.$
(4)

It can be seen from the following Theorem 4 and Proposition 1 that under
$H_{0}$, $T_{\operatorname{SUM}}/\hat{\sigma}_{S}$ has an asymptotically
standard normal distribution when both $n$ and $p$ go to infinity, where
$\hat{\sigma}_{S}^{2}\doteq\frac{2K}{n(n-1)}\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}^{2}$
and
$\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}\doteq\frac{1}{n(n-1)}\underset{t\not=s}{\sum\sum}(\bm{\varepsilon}_{t}^{\top}\bm{\varepsilon}_{s})^{2}$.
Hence, a level-$\alpha$ test will be performed by rejecting $H_{0}$ when
$T_{\operatorname{SUM}}/\hat{\sigma}_{S}$ is larger than $z_{\alpha}$, i.e.
the $1-\alpha$ quantile of the standard normal distribution.

Note that the test statistic in (4) is similar to the sum-type test statistic
proposed by Li et al. (2019). The differences between them are twofold: first,
the test statistic in (4) removes the diagonal elements
$\bm{\varepsilon}_{t}^{\top}\bm{\varepsilon}_{t}\bm{\varepsilon}_{t+l}^{\top}\bm{\varepsilon}_{t+l}$
from the summation to reduce the requirement of ${\bm{\Sigma}}$; second, we
use the martingale’s central limit theorem to establish the limiting null
distribution of the test statistic, while Li et al. (2019) used the random
matrix theory.

In deriving the asymptotic properties of $T_{\operatorname{SUM}}$, we impose
the following two conditions.

  * (C4)

Let $\bm{\varepsilon}_{t}={\bm{\Sigma}}^{1/2}{\bm{z}}_{t}$ under $H_{0}$,
where $\left\\{{{\bm{z}}}_{t}\right\\}$ with
${{\bm{z}}}_{t}=(z_{t1},\cdots,z_{tp})^{\top}$ is a sequence of
$p$-dimensional independent random vectors with independent components
$z_{ti}$’s, satisfying $\mathbb{E}z_{ti}=0$, $\mathbb{E}z_{ti}^{2}=1$ and
$\mathbb{E}z_{ti}^{4}<\infty$.

  * (C5)

$\mathrm{tr}({\bm{\Sigma}}^{4})=o\\{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\\}$.

######  Remark 3.

Condition (C5) is mild and holds automatically if all the eigenvalues of
${\bm{\Sigma}}$ are bounded, i.e. Condition (C5) is weaker than the condition
of bounded eigenvalues of ${\bm{\Sigma}}$ imposed in Li et al. (2019), which
indeed reduces the requirement of ${\bm{\Sigma}}$. Note that Condition (C5) is
also commonly adopted in the literature of testing high-dimensional covariance
matrices, such as in Chen et al. (2010).

######  Theorem 4.

Suppose Conditions (C4)-(C5) hold. Then, under $H_{0}$, we have
$T_{\operatorname{SUM}}/\sigma_{S}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1)$,
where
$\sigma_{S}^{2}\doteq\frac{2K}{n(n-1)}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})$.

Following the result in Proposition 1, we use the above
$\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}$ to estimate
$\mathrm{tr}({\bm{\Sigma}}^{2})$.

######  Proposition 1.

If $\bm{\varepsilon}_{t}={\bm{\Sigma}}^{1/2}{\bm{z}}_{t}$ and
$\mathrm{tr}({\bm{\Sigma}}^{4})=o\\{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\\}$,
then under $H_{0}$,
$\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}/\mathrm{tr}({\bm{\Sigma}}^{2})\mathop{\rightarrow}\limits^{p}1$.

Further, we present the asymptotic power function of the sum-type test
$\mathbb{I}(T_{\operatorname{SUM}}/\hat{\sigma}_{S}\geq z_{\alpha})$, when an
alternative hypothesis $H_{1}$ is specified. Here, we assume that under
$H_{1}$, the observations
$\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}\\}$ follow a
$p$-dimensional first-order vector moving average process, abbreviated as
VMA(1), of the form

$\displaystyle
H_{1}:\bm{\varepsilon}_{t}=\bm{A}_{0}{\bm{z}}_{t}+\bm{A}_{1}{\bm{z}}_{t-1},$
(5)

where $\bm{A}_{0},\bm{A}_{1}\in\mathbb{R}^{p\times p}$ are the coefficient
matrices. We consider the asymptotic distribution of $T_{\operatorname{SUM}}$
in the case of $K=1$ in the following Theorem 5.

######  Theorem 5.

Under $H_{1}$ in (5) with $K=1$, we have
$(T_{\operatorname{SUM}}-\mu_{S})/\sigma_{S1}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1)$,
where

$\displaystyle\mu_{S}\doteq$
$\displaystyle\mathrm{tr}(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1})+\frac{2}{T}\mathrm{tr}^{2}(\tilde{{\bm{\Sigma}}}_{01}),\quad\tilde{\bm{\Sigma}}_{0}\doteq{\bf
A}_{0}^{\top}{\bf A}_{0},\quad\tilde{\bm{\Sigma}}_{1}\doteq{\bf
A}_{1}^{\top}{\bf A}_{1},\quad\tilde{{\bm{\Sigma}}}_{01}\doteq{\bf
A}_{0}^{\top}{\bf A}_{1},$ $\displaystyle\sigma_{S1}^{2}$
$\displaystyle\doteq$
$\displaystyle\frac{2}{T^{2}}\mathrm{tr}^{2}(\tilde{\bm{\Sigma}}_{0}^{2}+\tilde{\bm{\Sigma}}_{1}^{2})+\frac{6}{T^{2}}\mathrm{tr}^{2}(\tilde{\bm{\Sigma}}_{0}\tilde{\bm{\Sigma}}_{1})$
$\displaystyle+\frac{4}{T}\left[2\operatorname{tr}(\widetilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1})^{2}+(\nu_{4}-3)\operatorname{tr}\left\\{D^{2}(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1})\right\\}\right]$
$\displaystyle+\frac{8}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top})\mathrm{tr}(\tilde{\bm{\Sigma}}_{0}^{2}+\tilde{\bm{\Sigma}}_{1}^{2})+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{1})\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{0})$
$\displaystyle+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{0}+\tilde{{\bm{\Sigma}}}_{1})\left\\{\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}^{\top}\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{0})+\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{01}^{\top}\tilde{\bm{\Sigma}}_{1})\right\\}$
$\displaystyle+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01})\left\\{\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{0}^{2}\tilde{{\bm{\Sigma}}}_{01}^{\top})+\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{1}^{2}\tilde{{\bm{\Sigma}}}_{01})+2\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0})\right\\}$
$\displaystyle+\frac{4}{T}\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0}^{2}+\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{1}^{2}+2\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0})$
$\displaystyle+\frac{4}{T}\operatorname{tr}(\widetilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01})+\frac{12}{T^{2}}\operatorname{tr}^{2}(\widetilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top})+\frac{16}{T^{2}}\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{01})\operatorname{tr}(\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}^{\top})$
$\displaystyle+\frac{4}{T^{2}}\operatorname{tr}^{2}(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{01})+\frac{4}{T^{2}}\operatorname{tr}^{2}(\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01})+r_{n},$

and the remainder $r_{n}=o(\sigma_{S1}^{2})$. Here, for each square matrix
$A$, $D(A)$ denotes the diagonal matrix consisting of the main diagonal
elements of $A$.

Similar to Proposition 1, we have
$\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}/\xi_{0}\mathop{\rightarrow}\limits^{p}1$
under $H_{1}$ in (5), where
$\xi_{0}\doteq\mathrm{tr}(\tilde{\bm{\Sigma}}_{0}^{2}+\tilde{\bm{\Sigma}}_{1}^{2})+2\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}^{\top}\tilde{\bm{\Sigma}}_{01})$.
As a result, the asymptotic power function of the proposed sum-type test
$\mathbb{I}(T_{\operatorname{SUM}}/\hat{\sigma}_{S}\geq z_{\alpha})$ under
$H_{1}$ in (5) is approximately equal to

$\displaystyle\beta_{\operatorname{SUM}}\doteq\Phi\left(\frac{\mu_{S}}{\sigma_{S1}}-z_{\alpha}\frac{\sqrt{2}n^{-1}\xi_{0}}{\sigma_{S1}}\right).$
(6)

###  2.4 Fisher’s combination test

To combine the proposed max-type and sum-type tests, we propose the Fisher’s
combination test, based on the asymptotic independence between
$T_{\operatorname{MAX}}$ and $T_{\operatorname{SUM}}$, to be provided in the
following Theorem 6. Specifically, let

$p_{\operatorname{MAX}}\doteq
1-G\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\right\\}$

and

$p_{\operatorname{SUM}}\doteq
1-\Phi\left(T_{\operatorname{SUM}}/\widehat{\sigma}_{S}\right)$

denote the $p$-values with respect to the test statistics
$T_{\operatorname{MAX}}$ and $T_{\operatorname{SUM}}$ respectively. Based on
$p_{\operatorname{MAX}}$ and $p_{\operatorname{SUM}}$, the proposed Fisher’s
combination test rejects $H_{0}$ at the significance level $\alpha$, if

$T_{\textrm{FC}}\doteq-2\log p_{\operatorname{MAX}}-2\log
p_{\operatorname{SUM}}$

is larger than $c_{\alpha}$, i.e. the $1-\alpha$ quantile of the chi-squared
distribution with 4 degrees of freedom (Fisher, 1950; Littell and Folks,
1971).

In deriving the asymptotic independence between $T_{\operatorname{MAX}}$ and
$T_{\operatorname{SUM}}$, we need to impose an additional condition as
follows. Let $\lambda_{\min}({\bm{\Sigma}})$ and
$\lambda_{\max}({\bm{\Sigma}})$ denote the minimum and maximum eigenvalues of
${\bm{\Sigma}}$, respectively.

  * (C6)

${{\mathrm{tr}^{-1}({\bm{\Sigma}}^{2})}(\log
p)^{\gamma}\max\\{\lambda_{\max}({\bm{\Sigma}})M_{p},M_{p}^{2},M_{p}^{3/2}\lambda_{\max}^{1/2}({\bm{\Sigma}})\\}}\to
0$ for some positive constant $\gamma>1$, where $M_{p}\doteq\max_{1\leq i\leq
p}\sum_{j\not=i}^{p}\sigma_{ij}^{2}$.

######  Remark 4.

Condition (C6) requires that the covariance between each pair of variables can
not be very large, which holds in many common cases. For example, it
automatically holds when all the variables are independent, i.e. $M_{p}=0$; if
all the eigenvalues of ${\bm{\Sigma}}$ are bounded, $M_{p}$ is also bounded,
hence Condition (C6) holds. In addition, if ${\bm{\Sigma}}$ is a bandable
covariance matrices, i.e. $\sigma_{ij}=0$ if $|i-j|>k$ for some fixed integer
$k$, and all nonzero $\sigma_{ij}$’s are bounded by $c$, then Condition (C6)
holds when $\lambda_{\max}({\bm{\Sigma}})(\log
p)^{\gamma}{\mathrm{tr}^{-1}({\bm{\Sigma}}^{2})}\to 0$.

Note that Condition (C6) is significantly different from Assumption 1 (ii) of
Yu et al. (2020) for testing high-dimensional covariance matrix. For example,
when $M_{p}=0$, we do not need to impose conditions on
$\lambda_{i}({\bm{\Sigma}})$’s. In fact, these two type of conditions cannot
contain each other.

######  Theorem 6.

Suppose $\bm{\varepsilon}_{t}\sim\mathcal{N}(\bm{0},{\bm{\Sigma}})$ for
$t=1,\cdots,n$ and Conditions (C2), (C3), (C5) and (C6) hold. Then, we have

$\displaystyle\operatorname{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leq
x,T_{\operatorname{SUM}}/\widehat{\sigma}_{S}\leq y\right\\}\rightarrow
G(x)\cdot\Phi(y),$ (7)

as $n,p\to\infty$, i.e. $T_{\operatorname{MAX}}$ and $T_{\operatorname{SUM}}$
are asymptotically independent.

Further, we relax the assumption of Gaussian distribution of
$\bm{\varepsilon}_{t}$ in Theorem 6 to non-Gaussian distributions, with sub-
gaussian-type or polynomial-type tails. To establish the theoretical result
under non-Gaussian distributions, Condition (C1) is modified as follows.

  * (C1′)

$\varepsilon_{ti}$’s have one of the following two types of tails: (i) sub-
gaussian-type tails, i.e. there exist some constant $\eta>0$ and $M>0$, such
that $\mathbb{E}e^{\eta\varepsilon_{ti}^{2}/\sigma_{i}^{2}}\leq M$ for all
$i\in\\{1,\cdots,p\\}$ and $t\in\\{1,\cdots,n\\}$, where $p$ satisfies $\log
p=o(n^{1/6})$; (ii) polynomial-type tails, i.e. for some $\gamma_{0}$ and
$c_{1}>0$, $p\leq c_{1}n^{\gamma_{0}}$ and for some $\epsilon>0$ and $M>0$,
$\mathbb{E}|\varepsilon_{ti}/\sigma_{i}|^{6\gamma_{0}+6+\epsilon}\leq M$ for
all $i\in\\{1,\cdots,p\\}$ and $t\in\\{1,\cdots,n\\}$.

######  Theorem 7.

Assume Conditions (C1′) and (C2)-(C6) hold. Then, we have

$\displaystyle\operatorname{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leq
x,T_{\operatorname{SUM}}/\widehat{\sigma}_{S}\leq y\right\\}\rightarrow
G(x)\cdot\Phi(y),$ (8)

as $n,p\to\infty$, i.e. $T_{\operatorname{MAX}}$ and $T_{\operatorname{SUM}}$
are asymptotically independent.

######  Remark 5.

Note that relaxing the assumption of Gaussian distribution in establishing the
asymptotically independence between the max-type and the sum-type statistics
is an important contribution of this paper, since all the existing literatures
on establishing such asymptotic independence, including Li and Xue (2015), Xu
et al. (2016), Feng et al. (2022) and Yu et al. (2020), are limited by the
assumption of Gaussian distribution. In this paper, we have developed a novel
theoretical tool to weaken the Gaussian distribution to non-Gaussian
distributions with sub-gaussian-type or polynomial-type tails. Its theoretical
framework is enlightening, which can be generalized to analogous studies.

Based on Theorem 6 or Theorem 7, we immediately have the following result for
$T_{\textrm{FC}}$.

######  Corollary 1.

Assume the same conditions as in Theorem 6 or Theorem 7, then we have
$T_{\textrm{FC}}\mathop{\rightarrow}\limits^{d}\chi_{4}^{2}$ as
$n,p\to\infty$.

Under the alternative hypothesis (3), we have $p_{{\rm MAX}}\to 0$ under the
sparse alternatives due to Theorem 2. On the other hand, under the dense
alternative hypothesis (5), we have $p_{{\rm SUM}}\to 0$ if
$\mu_{S}/\sigma_{S1}\to\infty$ due to Theorem 5.

According to the definition of $T_{\textrm{FCP}}$, if $p_{{\rm MAX}}\to 0$ or
$p_{{\rm SUM}}\to 0$, we have $T_{\textrm{FC}}\to\infty$, hence we reject the
null hypothesis.

######  Remark 6.

Note that Conditions (C2), (C3), (C5) and (C6) are all about ${\bm{\Sigma}}$,
which hold automatically if all the eigenvalues of ${\bm{\Sigma}}$ are
bounded. This indicates that these conditions are compatible and the
intersection of these conditions is very conventional, which means that the
scope of application of the proposed Fisher’s combination test is relatively
broad.

Next, we show that $T_{{\rm SUM}}$ is still asymptotically independent of
$T_{\rm MAX}$ under a specific alternative hypothesis. Based on this result,
we obtain a low bound of the power function of $T_{\rm FC}$.

######  Theorem 8.

Assume Conditions (C1′) and (C2)-(C5) hold. Assume that all eigenvalues of
${\bm{\Sigma}}=\mathrm{cov}(\bm{\varepsilon}_{t})$ are bounded. Let $K=1$.
Then, under the alternative hypothesis (5) with

${\bf A}_{0}=\left(\begin{array}[]{cc}{\bf A}_{011}&{\bf 0}\\\ {\bf 0}&{{\bf
A}_{022}}\end{array}\right),~{}{\bf A}_{1}=\left(\begin{array}[]{cc}{{\bf
A}}_{111}&{\bf 0}\\\ {\bf 0}&{\bf 0}\end{array}\right),$

${\bf A}_{011}\in\mathbb{R}^{d\times d}$, ${\bf
A}_{022}\in\mathbb{R}^{(p-d)\times(p-d)}$, ${\bf
A}_{111}\in\mathbb{R}^{d\times d}$ and $d=o(p)$, we have

$\displaystyle\operatorname{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leq
x,T_{\operatorname{SUM}}/\widehat{\sigma}_{S}\leq y\right\\}$
$\displaystyle\rightarrow\operatorname{P}\left\\{T_{\operatorname{MAX}}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leq
x\right\\}\operatorname{P}\left\\{T_{\operatorname{SUM}}/\widehat{\sigma}_{S}\leq
y\right\\},$ (9)

as $n,p\to\infty$, i.e. $T_{\operatorname{MAX}}$ and $T_{\operatorname{SUM}}$
are still asymptotically independent.

Define a minimal p-values test
$T_{\min}=\min(p_{\operatorname{SUM}},p_{\operatorname{MAX}})$. According to
Theorem 7, we reject the null hypothesis if $p_{\operatorname{SUM}}\leq
1-\sqrt{1-\alpha}\approx\alpha/2$ or $p_{\operatorname{MAX}}\leq
1-\sqrt{1-\alpha}\approx\alpha/2$. According to the results in Littell and
Folks (1971), we have that the power of Fisher’s combination test
$\beta_{T_{FC}}$ is comparable to the power of the minimal p-values test
$\beta_{T_{\min}}$. Thus, we have

$\displaystyle\beta_{T_{FC}}\approx\beta_{T_{\min}}\geq\beta_{T_{\operatorname{SUM}},\alpha/2}+\beta_{T_{\operatorname{MAX}},\alpha/2}-\beta_{T_{\operatorname{SUM}},\alpha/2}\beta_{T_{\operatorname{MAX}},\alpha/2},$

where the last inequality is based on the inclusion-exclusion principle and
the result of Theorem 8, and
$\beta_{T_{\operatorname{SUM}},\alpha},\beta_{T_{\operatorname{MAX}},\alpha},$
are respectively the power functions of the sum-type test
$T_{\operatorname{SUM}}$ and max-type test $T_{\operatorname{MAX}}$ at
significant level $\alpha$.

##  3 Numerical results

We now present some numerical results to demonstrate the performance of the
proposed max-type test, sum-type test and Fisher’s combined probability test,
abbreviated as MAX, SUM and FC respectively, as well as their comparison with
the sum-type test proposed by Li et al. (2019), abbreviated as LY. Note that
we exclude Chang, Yao and Zhou (2017) and Tsay (2020)’s max-type tests from
comparison because they have almost the same performance as the proposed max-
type test in the case of linear correlations.

###  3.1 Size performance

For the cases under the null hypothesis, we let $\bm{\varepsilon}_{t}={\bf
A}{\bm{z}}_{t}$ with ${\bm{z}}_{t}=(z_{t1},\cdots,z_{tp})^{\top}$ and ${\bf
A}=\\{a_{ij}\\}_{1\leq i,j\leq p}$. We consider the following two
distributions of ${\bm{z}}_{t}$: (i) ${\bm{z}}_{t}\stackrel{{\scriptstyle
i.i.d}}{{\sim}}\mathcal{N}(\bm{0},{\bf I}_{p})$; (ii)
$z_{ti}\stackrel{{\scriptstyle i.i.d}}{{\sim}}Ga(4,0.5)-2$, and the following
three settings of ${\bf A}$:

  * (I)

${\bf A}={\bm{\Sigma}}^{1/2}$, ${\bm{\Sigma}}=\\{\sigma_{ij}\\}_{1\leqslant
i,j\leqslant p}$, $\sigma_{ii}=1$, $i=1,\cdots,p$, $\sigma_{ij}=0.5(i-j)^{-2}$
with $i\neq j$;

  * (II)

${\bf A}={\bm{\Sigma}}^{1/2}$, ${\bm{\Sigma}}=\\{\sigma_{ij}\\}_{1\leqslant
i,j\leqslant p}$, $\sigma_{ii}=1$, $i=1,\cdots,p$,
$\sigma_{ij}=0.5\mathbb{I}(|i-j|<5)$ with $i\neq j$;

  * (III)

$a_{ij}\stackrel{{\scriptstyle i.i.d}}{{\sim}}U(-1,1)$.

Tables 1 and 2 summarize the empirical size performance of MAX, LY, SUM and FC
under settings (I)-(III) with the distributions (i) and (ii), respectively.
Both of these tables suggest that in terms of size performance, SUM is the
best and MAX is the most conservative, while FC and LY are in between. In
addition, we find that FC has much better size performance than LY in
situations where $p/n$ is relatively large or $K>1$.

Table 1:  Size performance in the case of $\bm{\varepsilon}_{t}={\bf A}{\bm{z}}_{t}$ with distribution (i): ${\bm{z}}_{t}\stackrel{{\scriptstyle i.i.d}}{{\sim}}\mathcal{N}(\bm{0},{\bf I}_{p})$. |  | $K=1$ | $K=2$ | $K=3$
---|---|---|---|---
$n$ | $p$ | MAX | LY | SUM | FC | MAX | LY | SUM | FC | MAX | LY | SUM | FC
|  | Setting (I)
100 | 30 | 0.016 | 0.039 | 0.046 | 0.043 | 0.010 | 0.025 | 0.043 | 0.025 | 0.016 | 0.026 | 0.068 | 0.050
100 | 60 | 0.007 | 0.036 | 0.050 | 0.030 | 0.004 | 0.012 | 0.042 | 0.021 | 0.006 | 0.004 | 0.043 | 0.024
100 | 90 | 0.004 | 0.022 | 0.048 | 0.024 | 0.010 | 0.012 | 0.043 | 0.025 | 0.006 | 0.002 | 0.047 | 0.015
100 | 120 | 0.008 | 0.016 | 0.043 | 0.021 | 0.006 | 0.004 | 0.040 | 0.016 | 0.005 | 0.000 | 0.048 | 0.025
200 | 30 | 0.012 | 0.054 | 0.062 | 0.052 | 0.020 | 0.044 | 0.052 | 0.053 | 0.014 | 0.040 | 0.053 | 0.038
200 | 60 | 0.017 | 0.048 | 0.056 | 0.039 | 0.016 | 0.035 | 0.049 | 0.032 | 0.020 | 0.032 | 0.056 | 0.047
200 | 90 | 0.016 | 0.032 | 0.049 | 0.027 | 0.011 | 0.030 | 0.054 | 0.037 | 0.017 | 0.021 | 0.056 | 0.050
200 | 120 | 0.014 | 0.031 | 0.049 | 0.036 | 0.017 | 0.016 | 0.043 | 0.036 | 0.012 | 0.015 | 0.065 | 0.039
|  | Setting (II)
100 | 30 | 0.014 | 0.038 | 0.050 | 0.043 | 0.009 | 0.037 | 0.052 | 0.038 | 0.005 | 0.022 | 0.051 | 0.032
100 | 60 | 0.003 | 0.030 | 0.049 | 0.021 | 0.010 | 0.017 | 0.050 | 0.025 | 0.009 | 0.011 | 0.049 | 0.025
100 | 90 | 0.008 | 0.027 | 0.045 | 0.031 | 0.011 | 0.011 | 0.056 | 0.040 | 0.006 | 0.007 | 0.046 | 0.021
100 | 120 | 0.008 | 0.017 | 0.042 | 0.028 | 0.002 | 0.008 | 0.052 | 0.022 | 0.004 | 0.001 | 0.035 | 0.022
200 | 30 | 0.018 | 0.045 | 0.053 | 0.043 | 0.014 | 0.045 | 0.051 | 0.043 | 0.010 | 0.030 | 0.048 | 0.035
200 | 60 | 0.016 | 0.041 | 0.052 | 0.033 | 0.016 | 0.031 | 0.056 | 0.039 | 0.013 | 0.017 | 0.047 | 0.027
200 | 90 | 0.016 | 0.028 | 0.040 | 0.023 | 0.017 | 0.023 | 0.049 | 0.043 | 0.017 | 0.009 | 0.050 | 0.031
200 | 120 | 0.015 | 0.035 | 0.050 | 0.037 | 0.011 | 0.013 | 0.043 | 0.028 | 0.011 | 0.008 | 0.052 | 0.030
|  | Setting (III)
100 | 30 | 0.014 | 0.038 | 0.050 | 0.043 | 0.009 | 0.037 | 0.052 | 0.038 | 0.005 | 0.022 | 0.051 | 0.032
100 | 60 | 0.003 | 0.030 | 0.049 | 0.021 | 0.010 | 0.017 | 0.050 | 0.025 | 0.009 | 0.011 | 0.049 | 0.025
100 | 90 | 0.008 | 0.027 | 0.045 | 0.031 | 0.011 | 0.011 | 0.056 | 0.040 | 0.006 | 0.007 | 0.046 | 0.021
100 | 120 | 0.008 | 0.017 | 0.042 | 0.028 | 0.002 | 0.008 | 0.052 | 0.022 | 0.004 | 0.001 | 0.035 | 0.022
200 | 30 | 0.018 | 0.045 | 0.053 | 0.043 | 0.014 | 0.045 | 0.051 | 0.043 | 0.010 | 0.030 | 0.048 | 0.035
200 | 60 | 0.016 | 0.041 | 0.052 | 0.033 | 0.016 | 0.031 | 0.056 | 0.039 | 0.013 | 0.017 | 0.047 | 0.027
200 | 90 | 0.016 | 0.028 | 0.040 | 0.023 | 0.017 | 0.023 | 0.049 | 0.043 | 0.017 | 0.009 | 0.050 | 0.031
200 | 120 | 0.015 | 0.035 | 0.050 | 0.037 | 0.011 | 0.013 | 0.043 | 0.028 | 0.011 | 0.008 | 0.052 | 0.030
Table 2:  Size performance in the case of $\bm{\varepsilon}_{t}={\bf A}{\bm{z}}_{t}$ with distribution (ii): $z_{ti}\stackrel{{\scriptstyle i.i.d}}{{\sim}}Ga(4,0.5)-2$. |  | $K=1$ | $K=2$ | $K=3$
---|---|---|---|---
$n$ | $p$ | MAX | LY | SUM | FC | MAX | LY | SUM | FC | MAX | LY | SUM | FC
|  | Setting (I)
100 | 30 | 0.011 | 0.050 | 0.058 | 0.049 | 0.008 | 0.038 | 0.060 | 0.036 | 0.012 | 0.018 | 0.053 | 0.031
100 | 60 | 0.010 | 0.023 | 0.043 | 0.027 | 0.009 | 0.014 | 0.034 | 0.020 | 0.009 | 0.007 | 0.040 | 0.020
100 | 90 | 0.006 | 0.026 | 0.044 | 0.022 | 0.006 | 0.013 | 0.058 | 0.031 | 0.008 | 0.002 | 0.058 | 0.027
100 | 120 | 0.011 | 0.017 | 0.046 | 0.028 | 0.006 | 0.012 | 0.046 | 0.019 | 0.009 | 0.001 | 0.047 | 0.022
200 | 30 | 0.018 | 0.046 | 0.052 | 0.054 | 0.012 | 0.047 | 0.064 | 0.041 | 0.013 | 0.027 | 0.039 | 0.031
200 | 60 | 0.014 | 0.047 | 0.060 | 0.037 | 0.013 | 0.038 | 0.054 | 0.035 | 0.012 | 0.029 | 0.045 | 0.041
200 | 90 | 0.015 | 0.034 | 0.049 | 0.039 | 0.015 | 0.026 | 0.055 | 0.035 | 0.010 | 0.009 | 0.042 | 0.025
200 | 120 | 0.022 | 0.040 | 0.059 | 0.038 | 0.014 | 0.010 | 0.043 | 0.022 | 0.016 | 0.010 | 0.057 | 0.029
|  | Setting (II)
100 | 30 | 0.010 | 0.049 | 0.061 | 0.048 | 0.003 | 0.032 | 0.053 | 0.035 | 0.011 | 0.021 | 0.044 | 0.032
100 | 60 | 0.012 | 0.023 | 0.042 | 0.029 | 0.013 | 0.022 | 0.053 | 0.038 | 0.008 | 0.016 | 0.063 | 0.037
100 | 90 | 0.007 | 0.029 | 0.053 | 0.028 | 0.008 | 0.008 | 0.051 | 0.024 | 0.003 | 0.002 | 0.055 | 0.031
100 | 120 | 0.005 | 0.017 | 0.052 | 0.025 | 0.005 | 0.006 | 0.040 | 0.016 | 0.009 | 0.002 | 0.051 | 0.023
200 | 30 | 0.016 | 0.045 | 0.057 | 0.044 | 0.019 | 0.045 | 0.059 | 0.045 | 0.019 | 0.044 | 0.060 | 0.051
200 | 60 | 0.013 | 0.042 | 0.055 | 0.038 | 0.007 | 0.023 | 0.044 | 0.030 | 0.015 | 0.019 | 0.050 | 0.029
200 | 90 | 0.015 | 0.033 | 0.046 | 0.037 | 0.013 | 0.027 | 0.070 | 0.040 | 0.016 | 0.015 | 0.049 | 0.031
200 | 120 | 0.009 | 0.024 | 0.048 | 0.029 | 0.018 | 0.027 | 0.050 | 0.042 | 0.008 | 0.013 | 0.053 | 0.028
|  | Setting (III)
100 | 30 | 0.010 | 0.049 | 0.061 | 0.048 | 0.003 | 0.032 | 0.053 | 0.035 | 0.011 | 0.021 | 0.044 | 0.032
100 | 60 | 0.012 | 0.023 | 0.042 | 0.029 | 0.013 | 0.022 | 0.053 | 0.038 | 0.008 | 0.016 | 0.063 | 0.037
100 | 90 | 0.007 | 0.029 | 0.053 | 0.028 | 0.008 | 0.008 | 0.051 | 0.024 | 0.003 | 0.002 | 0.055 | 0.031
100 | 120 | 0.005 | 0.017 | 0.052 | 0.025 | 0.005 | 0.006 | 0.040 | 0.016 | 0.009 | 0.002 | 0.051 | 0.023
200 | 30 | 0.016 | 0.045 | 0.057 | 0.044 | 0.019 | 0.045 | 0.059 | 0.045 | 0.019 | 0.044 | 0.060 | 0.051
200 | 60 | 0.013 | 0.042 | 0.055 | 0.038 | 0.007 | 0.023 | 0.044 | 0.030 | 0.015 | 0.019 | 0.050 | 0.029
200 | 90 | 0.015 | 0.033 | 0.046 | 0.037 | 0.013 | 0.027 | 0.070 | 0.040 | 0.016 | 0.015 | 0.049 | 0.031
200 | 120 | 0.009 | 0.024 | 0.048 | 0.029 | 0.018 | 0.027 | 0.050 | 0.042 | 0.008 | 0.013 | 0.053 | 0.028

###  3.2 Power comparison

In this subsection, we compare the empirical power performance of the above
four tests. For the cases under the alternative hypothesis, we only consider
the above distribution (i) to avoid redundancy and consider the following
three new settings of $\bm{\varepsilon}_{t}$:

  * (IV)

VAR(1) model: $\bm{\varepsilon}_{t}={\bf
A}\bm{\varepsilon}_{t-1}+{\bm{z}}_{t}$;

  * (V)

VMA(1) model: $\bm{\varepsilon}_{t}={\bm{z}}_{t}+{\bf A}{\bm{z}}_{t-1}$;

  * (VI)

VARMA(1) model: $\bm{\varepsilon}_{t}=0.5{\bf
A}\bm{\varepsilon}_{t-1}+{\bm{z}}_{t}+0.5{\bf A}{\bm{z}}_{t-1}$.

Here “VAR(1)”, “VMA(1)” and “VARMA(1)” are the abbreviations of 1-order vector
autoregressive process, vector moving average process and vector
autoregressive moving average process, respectively. Let ${\bf
A}=\\{a_{ij}\\}_{1\leq i,j\leq p}$. For the alternative hypothesis, we let the
first $a_{ij}\not=0$ for $1\leq i,j\leq m$ and $a_{ij}=0$ otherwise. Note that
$m$ controls the signal strength and sparsity of ${\bf A}$. For the VAR(1)
model, if $m=1$, $a_{ij}\sim U(0.4,0.8)$; if $2\leq m\leq 10$, $a_{ij}\sim
U(-1.4/m,1.4/m)$. For the VMA(1) model, if $m=1$, $a_{ij}\sim U(0.4,0.9)$; if
$2\leq m\leq 10$, $a_{ij}\sim U(-1.8/m,1.8/m)$. For the VARMA(1) model, if
$m=1$, $a_{ij}\sim U(0.4,0.8)$; if $2\leq m\leq 10$, $a_{ij}\sim
U(-1.6/m,1.6/m)$. Specifically, as $m$ decreases, both the signal strength and
sparsity of ${\bf A}$ increase. Let $n=200$, $p\in\\{60,90\\}$ and
$K\in\\{1,2,3\\}$.

Figure 1: Power curves of the involved tests with $m=1,2,\cdots,10$ and
$(n,p)=(200,60)$. Figure 2: Power curves of the involved tests with
$m=1,2,\cdots,10$ and $(n,p)=(200,90)$.

Figures 1 and 2 present the empirical power curves of MAX, LY, SUM and FC
under settings (IV)-(VI) and distribution (i) for $(n,p)=(200,60)$ and
$(200,90)$, respectively. In each panel of these figures, the abscissa $m$
varies between 1 and 10, corresponding to the power performance of the
involved tests with different signal strength and sparsity of ${\bf A}$. Both
figures suggest that in terms of empirical power performance, FC is better
than its competitors in most cases whether ${\bf A}$ is sparse or non-sparse,
which has robust performance due to the combination of the advantages of both
MAX and SUM. Although FC cannot outperform its competitors in all cases, it is
indeed applicable to both sparse and non-sparse cases of ${\bf A}$. In
contrast, MAX generally fails when ${\bf A}$ is dense enough, while SUM and LY
generally fail when ${\bf A}$ is very sparse.

Note that in all the above simulation studies, the ways of setting up
${\bm{\Sigma}}$ are very common, all of which satisfy the condition of bounded
eigenvalues of ${\bm{\Sigma}}$. Hence, these simulation settings for
${\bm{\Sigma}}$ satisfy all the conditions imposed on ${\bm{\Sigma}}$ in the
theoretical results established above.

##  4 Application

In this section, we are interested in testing whether the error series
$\\{\bm{\varepsilon}_{t}\\}$ under the Fama-French three-factor model (Fama
and French, 1993) is white noise, i.e.

$\displaystyle H_{0}:\\{\bm{\varepsilon}_{t}\\}\mbox{~{}is white noise
versus}~{}H_{1}:\\{\bm{\varepsilon}_{t}\\}\mbox{~{}is not white noise},$ (10)

where $\bm{\varepsilon}_{t}=(\varepsilon_{t1},\cdots,\varepsilon_{tp})^{\top}$
and $p$ is the number of securities. The Fama-French three-factor model is one
of the most popular factor pricing models in finance, which has the explicit
form

$Y_{ti}=r_{ti}-r_{ft}=\alpha_{i}+\beta_{i1}(r_{mt}-r_{ft})+\beta_{i2}SMB_{t}+\beta_{i3}HML_{t}+\varepsilon_{ti}$

for $t\in\\{1,\cdots,n\\}$ and $i\in\\{1,\cdots,p\\}$, where $r_{ti}$ is the
return of the $i$-th security at time $t$, $r_{ft}$ is the risk free rate at
time $t$, $Y_{ti}=r_{ti}-r_{ft}$ is the excess return of the $i$-th security
at time $t$ and $r_{mt}$ is the market return at time $t$.

We collected the return data of the securities in the S&P 500 index and
considered two forms of data compilation. First, we compiled the monthly
returns on all the securities that constitute the S&P 500 index each month
over the period from January 2005 to November 2018. Because the securities
that make up the index change over time, we only consider $p=374$ securities
that were included in the S&P 500 index during the entire period. A total of
$T=165$ consecutive observations were obtained. The time series data on the
safe rate of return, and the market factors are obtained from Ken French’s
data library web page. The one-month US treasury bill rate is chosen as the
risk-free rate ($r_{ft}$). The value-weighted return on all NYSE, AMEX, and
NASDAQ stocks from CRSP is used as a proxy for the market return ($r_{mt}$).
The average return on the three small portfolios minus the average return on
the three big portfolios ($SMB_{t}$), and the average return on two value
portfolios minus the average return on two growth portfolios ($HML_{t}$) are
calculated based on the stocks listed on the NYSE, AMEX and NASDAQ.

Second, we compiled the weekly returns on all the securities that constitute
the S&P 500 index over the period from January 2005 to November 2018. The
weekly data were calculated using the security prices on Fridays. Similar to
the monthly data, we only considered a total of $p=381$ stocks that were
included in the S&P 500 index during the entire period. We formed a total of
$T=716$ weekly return rates for each stock during this period after excluding
the Fridays that happened to be holidays.

Under these two forms of data compilation, we test the hypotheses in (10)
using the proposed Fisher’s combination test as well as its competitors,
respectively. Specifically, we let

$\hat{\varepsilon}_{ti}\doteq
Y_{ti}-\hat{\alpha}_{i}-\hat{\beta}_{i1}(r_{mt}-r_{ft})-\hat{\beta}_{i2}SMB_{t}-\hat{\beta}_{i3}HML_{t},$

where $\hat{\alpha}_{i}$, $\hat{\beta}_{i1}$, $\hat{\beta}_{i2}$ and
$\hat{\beta}_{i3}$ are the ordinary least squares (OLS) estimators of
${\alpha}_{i}$, ${\beta}_{i1}$, ${\beta}_{i2}$ and ${\beta}_{i3}$,
respectively. To demonstrate the usefulness of the proposed test, we treat
$\hat{\bm{\varepsilon}}_{t}=(\hat{\varepsilon}_{t1},\cdots,\hat{\varepsilon}_{tp})^{\top}$
as the observation of $\bm{\varepsilon}_{t}$, instead of considering the
testing problem within the Fama-French three-factor model.

We use the sliding window method for the subsequent application. Given a fixed
length $n$, for each $\tau\in\\{1,\cdots,T-n\\}$, we implement each of the
involved tests on the data compiled from the period from $\tau$ to $\tau+n-1$,
where $\\{\tau,\cdots,\tau+n-1\\}$ is the sliding window of length $n$. Then,
we record the rate of rejecting the null hypothesis in these $T-n$ testing
results corresponding to the $T-n$ sliding windows.

Due to the great complexity and diversity of the financial market, the Fama-
French three-factor model is only an approximation and the three included
factors may often fail to accurately describe the generating mechanism of the
excess returns of a large number of securities. Nevertheless, it has played an
important role in pricing analysis of securities. This certainly motivates the
investigation on whether a certain factor pricing model is sufficient and
whether more advanced factor pricing models with more explanatory factors are
needed. It is not irrational to suspect that the Fama-French three-factor
model is not sufficient hence the null hypothesis may not be true, especially
in the high-dimensional situations. To this end, of course a testing method
with more tendency of rejection may be considered to perform better, as long
as the test can control the effect size.

Table 3 summarizes the rejecting rates for each $n\in\\{40,50,60,70\\}$ and
each form of data compilation, where the prescribed integer $K\in\\{2,3\\}$ is
used to establish the proposed test statistics. It suggests that for the
weekly data, FC, MAX and SUM are more inclined to reject the null hypothesis
than LY, where FC is the most powerful and MAX is the second. This may be due
to the stronger dependence of securities on time series of weekly data,
compared with the monthly data, which may lead to some correlation matrices
with larger signal strength. In such a circumstance, both the max-type and
sum-type of tests can perform well, and the Fisher’s combined probability test
FC outperform them as a combination of them.

Compared with weekly time series, the time dependence of monthly time series
is much weaker, which leads to some correlation matrices with much weaker
signal strength. This may be the reason why MAX fails to deal with the monthly
data, while SUM, FC and LY have good performance. In particular, SUM
outperform all the remaining methods in such circumstance.

Overall, SUM, LY and MAX can only have good performance in their respective
suitable situations, while FC can have robust performance in both situations.

Table 3:  Rejecting rates for the weekly and monthly data respectively. |  | $K=2$ | $K=3$
---|---|---|---
$n$ | $p$ | MAX | LY | SUM | FC | MAX | LY | SUM | FC
|  | Weekly data
40 | 381 | 0.74 | 0.19 | 0.83 | 0.93 | 0.72 | 0.24 | 0.63 | 0.86
50 | 381 | 0.86 | 0.26 | 0.79 | 0.98 | 0.83 | 0.32 | 0.63 | 0.94
60 | 381 | 0.88 | 0.34 | 0.73 | 0.98 | 0.87 | 0.39 | 0.57 | 0.97
70 | 381 | 0.89 | 0.46 | 0.58 | 0.98 | 0.90 | 0.41 | 0.64 | 0.99
|  | Monthly data
40 | 374 | 0.00 | 0.45 | 0.64 | 0.36 | 0.00 | 0.45 | 0.86 | 0.67
50 | 374 | 0.00 | 0.55 | 0.68 | 0.55 | 0.01 | 0.62 | 0.90 | 0.75
60 | 374 | 0.00 | 0.50 | 0.70 | 0.49 | 0.01 | 0.59 | 0.95 | 0.84
70 | 374 | 0.21 | 0.56 | 0.69 | 0.53 | 0.18 | 0.69 | 0.99 | 0.93

##  5 Conclusion

Driven by the task of testing for high-dimensional white noise, we adopt the
strategy of combining independent tests of hypotheses. In particular, we
employ the well known Fisher’s combination test to combine the max-type and
sum-type statistics, which is guaranteed to be valid by the asymptotic
independence between the two types of statistics. Through extensive numerical
results, we demonstrate that the proposed test has clear advantages in power
comparison, due to its robustness to sparsity of the serial correlation
structure. Furthermore, via an empirical application, we demonstrate the
robust performance of the proposed test in testing white noise of the return
data of the S$\&$P 500 securities under the Fama-French three-factor model.

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Supplementary Material of “Testing for high-dimensional white noise”

##  6 Technical proofs

###  6.1 Proof of Theorem 1

First, we present some technical results for the proof of Theorem 1.

We restate Theorem 2 in Feng et al. (2022) as the following Proposition 2, in
which the following condition is imposed.

  * (CA1)

Let ${\bm{\Sigma}}=\\{\sigma_{ij}\\}_{1\leq i,j\leq p}$. For some
$\varrho\in(0,1)$, assume $|\sigma_{ij}|\leq\varrho$ for all $1\leq i<j\leq p$
and $p\geq 2$. Suppose $\\{\delta_{p}:p\geq 1\\}$ and $\\{\kappa_{p}:p\geq
1\\}$ are positive constants with $\delta_{p}=o(1/\log p)$ and
$\kappa=\kappa_{p}\to 0$ as $p\to\infty$. For $1\leq i\leq p$, define
$B_{p,i}=\big{\\{}1\leq j\leq p:|\sigma_{ij}|\geq\delta_{p}\big{\\}}$ and
$C_{p}=\big{\\{}1\leq i\leq p:|B_{p,i}|\geq p^{\kappa}\big{\\}}$. Assume that
$|C_{p}|/p\to 0$ as $p\to\infty$.

######  Proposition 2.

Suppose $(Z_{1},\cdots,Z_{p})^{\top}\sim\mathcal{N}(\bm{0},{\bm{\Sigma}})$ and
Condition (CA1) holds. Then, we have $\max_{1\leq i\leq p}Z_{i}^{2}-2\log
p+\log\log p$ converges to a Gumbel distribution with cdf
$G(x)=\exp(-\frac{1}{\sqrt{\pi}}e^{-x/2})$ as $p\to\infty$.

The following lemma is from the proof of Theorem 2 in Feng et al. (2022).

######  Lemma 1.

Suppose $(Z_{1},\cdots,Z_{p})^{\top}\sim\mathcal{N}(\bm{0},{\bm{\Sigma}})$ and
Condition (CA1) holds. For any $x\in\mathbb{R}$ and any $1\leq t\leq p$, let

$\displaystyle\alpha_{t}=\sum\operatorname{P}(|Z_{i_{1}}|>z,\cdots,|Z_{i_{t}}|>z),~{}z=\big{(}2\log
p-\log\log p+x\big{)}^{1/2},$

where the sum runs over all $i_{1}<\cdots<i_{t}$ with $i_{1}\cdots,i_{t}\in
D_{p}=\\{1,\cdots,p\\}\setminus C_{p}$. Then,

$\displaystyle\lim_{p\to\infty}\alpha_{t}=\frac{1}{t!}\pi^{-t/2}e^{-tx/2}.$
(11)

######  Lemma 2.

(Bernstein’s inequality) Let $X_{1},\ldots,X_{n}$ be independent centered
random variables a.s. bounded by $A<\infty$ in absolute value. Let
$\sigma^{2}=n^{-1}\sum_{i=1}^{n}\mathbb{E}\left(X_{i}^{2}\right).$ Then for
all $x>0$,

$\operatorname{P}\left(\sum_{i=1}^{n}X_{i}\geq
x\right)\leq\exp\left(-\frac{x^{2}}{2n\sigma^{2}+2Ax/3}\right).$

Define $\hat{\sigma}_{i}^{2}=\frac{1}{n}\sum_{t=1}^{n}\varepsilon_{ti}^{2}$
and $\sigma_{i}^{2}=\mathrm{var}(\varepsilon_{ti})$.

######  Lemma 3.

Suppose Condition (C2) holds. Then, under $H_{0}$,

  * (1)

if (C1)-(i) holds, we have

$\displaystyle\operatorname{P}\left(\max_{1\leq i\leq
p}\left|\hat{\sigma}_{i}^{2}-\sigma_{i}^{2}\right|\geq
C\frac{\epsilon_{n}}{\log p}\right)=O(p^{-1}),$ (12)

  * (2)

if (C1)-(ii) holds,

$\displaystyle\operatorname{P}\left(\max_{1\leq i\leq
p}\left|\hat{\sigma}_{i}^{2}-\sigma_{i}^{2}\right|\geq
C\frac{\epsilon_{n}}{\log p}\right)=O(n^{-\epsilon/8}),$ (13)

as $\epsilon_{n}\doteq\max\left\\{(\log p)^{1/6}/n^{1/2},(\log
p)^{-1}\right\\}\rightarrow 0$.

Proof. We first assume that (C1)-(i) holds. It suffices to show that, for any
$\delta>0$,

$\displaystyle\mathrm{P}\left\\{\max_{1\leq i\leq
p}\left|\frac{1}{n}\sum_{k=1}^{n}\left(\varepsilon_{ki}^{2}-\mathbb{E}\varepsilon_{ki}^{2}\right)\right|\geq
C\sqrt{\frac{\log p}{n}}\right\\}=O\left(p^{-{\delta}}\right).$ (14)

Define

$\tilde{\varepsilon}_{ki}\doteq\varepsilon_{ki}\mathbb{I}\left\\{\left|\varepsilon_{ki}\right|\leq\tau\sqrt{\log(p+n)}\right\\},$

where $\tau$ is sufficiently large. We have

$\displaystyle\left|\mathbb{E}\varepsilon_{ki}^{2}-\mathbb{E}\tilde{\varepsilon}_{ki}^{2}\right|$
$\displaystyle\leq
C\left(\mathbb{E}\varepsilon_{ki}^{4}\mathbb{E}\left[I\left\\{\left|\varepsilon_{ki}\right|\geq\tau\sqrt{\log(p+n)}\right\\}\right]\right)^{1/2}$
$\displaystyle\leq
C(n+p)^{-\tau^{2}\eta/2}\left\\{\mathbb{E}\varepsilon_{ki}^{4}\exp\left(2^{-1}\eta\varepsilon_{ki}^{2}\right)\right\\}^{1/2}$
$\displaystyle\leq C(n+p)^{-\tau^{2}\eta/2},$ (15)

where $C$ does not depend on $n$ and $p$. Thus, it follows that

$\displaystyle\mathrm{P}\left\\{\max_{1\leq i\leq
p}\left|\frac{1}{n}\sum_{k=1}^{n}\left(\varepsilon_{ki}^{2}-\mathbb{E}\varepsilon_{ki}^{2}\right)\right|\geq
C\sqrt{\frac{\log p}{n}}\right\\}$ $\displaystyle\leq$
$\displaystyle\mathrm{P}\left\\{\max_{1\leq i\leq
p}\left|\frac{1}{n}\sum_{k=1}^{n}\left(\tilde{\varepsilon}_{ki}^{2}-\mathbb{E}\tilde{\varepsilon}_{ki}^{2}\right)\right|\geq
2^{-1}C\sqrt{\left.\frac{\log p}{n}\right)}\right\\}$
$\displaystyle+np\mathrm{P}\left\\{\left|\varepsilon_{ki}\right|\geq\tau\sqrt{\log(p+n)}\right\\},$

where

$np\mathrm{P}\left(\left|\varepsilon_{ki}\right|\geq\tau\sqrt{\log(p+n)}\right)\leq
np(n+p)^{-\tau^{2}\eta}\mathbb{E}\exp\left(\eta\varepsilon_{ki}^{2}\right)=O\left(p^{-{\delta}}\right).$

Let $t=\eta\left(8\tau^{2}\right)^{-1}\sqrt{\log p/n}$ and
$\tilde{Z}_{ki}=\tilde{\varepsilon}_{ki}^{2}-\mathbb{E}\tilde{\varepsilon}_{ki}^{2}$.
Then, we have

$\displaystyle\mathrm{P}\left\\{\frac{1}{n}\sum_{k=1}^{n}\left(\tilde{\varepsilon}_{ki}^{2}-\mathbb{E}\tilde{\varepsilon}_{ki}^{2}\right)\geq
C\sqrt{\frac{\log p}{n}}\right\\}$ $\displaystyle\leq$
$\displaystyle\exp\left(-Ct\sqrt{n\log
p}\right)\prod_{k=1}^{n}\mathbb{E}\exp\left(t\tilde{Z}_{ki}\right)$
$\displaystyle\leq$ $\displaystyle\exp\left(-Ct\sqrt{n\log
p}\right)\prod_{k=1}^{n}\left\\{1+\mathbb{E}t^{2}\tilde{Z}_{ki}^{2}\exp\left(t\left|\tilde{Z}_{ki}\right|\right)\right\\}$
$\displaystyle\leq$ $\displaystyle\exp\left\\{-Ct\sqrt{n\log
p}+\sum_{k=1}^{n}\mathbb{E}t^{2}\tilde{Z}_{ki}^{2}\exp\left(t\left|\tilde{Z}_{ki}\right|\right)\right\\}$
$\displaystyle\leq$
$\displaystyle\exp\left\\{-C\eta\left(8\tau^{2}\right)^{-1}\log
p+c_{\tau,\eta}\log p\right\\}\leq Cp^{-{\delta}},$

where $c_{\tau,\eta}$ is a positive constant depending only on $\tau$ and
$\eta$. Similarly, we can show that

$\mathrm{P}\left\\{\frac{1}{n}\sum_{k=1}^{n}\left(\tilde{\varepsilon}_{ki}^{2}-\mathbb{E}\tilde{\varepsilon}_{ki}^{2}\right)\leq-C\sqrt{\frac{\log
p}{n}}\right\\}\leq Cp^{-{\delta}},$

which leads to (14).

It remains to prove this lemma under (C2)-(ii). Define

$\hat{\varepsilon}_{ki}^{2}=\varepsilon_{ki}^{2}\mathbb{I}\left\\{\left|\varepsilon_{ki}^{2}\right|\leq
n/(\log p)^{8}\right\\}.$

Then, as in $(\ref{s6})$, we can show that
$\left|\mathbb{E}\varepsilon_{ki}^{2}-\mathbb{E}\hat{\varepsilon}_{ki}^{2}\right|\leq
Cn^{-\gamma_{0}/4}.$ It follows that

$\displaystyle\mathrm{P}\left(\max_{1\leq i\leq
p}\left|\sum_{k=1}^{n}\left(\varepsilon_{ki}^{2}-\mathbb{E}\varepsilon_{ki}^{2}\right)\right|\geq\frac{n\epsilon_{n}}{\log
p}\right)$ $\displaystyle\leq$ $\displaystyle\mathrm{P}\left(\max_{1\leq i\leq
p}\left|\sum_{k=1}^{n}\left(\hat{\varepsilon}_{ki}^{2}-\mathbb{E}\hat{\varepsilon}_{ki}^{2}\right)\right|\geq
2^{-1}\frac{n\epsilon_{n}}{\log
p}\right)+\mathrm{P}\left(\max_{i,k}\left|\varepsilon_{ki}^{2}\right|\geq\frac{n}{(\log
p)^{8}}\right)$ $\displaystyle\leq$ $\displaystyle Cp^{2}\exp\left\\{-C(\log
p)^{4}\right\\}+Cn^{-\epsilon/8},$

where the last inequality follows from Lemma 2 and (C2)-(ii). Then, the proof
of this lemma is completed. $\square$

Define $\tilde{\Gamma}(k)=\left\\{\tilde{\rho}_{ij}(k)\right\\}_{1\leqslant
i,j\leqslant
p}\doteq\operatorname{diag}\\{{\bm{\Sigma}}(0)\\}^{-1/2}\hat{\bm{\Sigma}}(k)\operatorname{diag}\\{{\bm{\Sigma}}(0)\\}^{-1/2}.$
We have the following results for $\tilde{\rho}_{ij}(k)$.

######  Lemma 4.

Under $H_{0}$, we have

  * (1)

if (C1)-(i) holds, we have

$\displaystyle\operatorname{P}\left\\{\max_{(i,j,k)\in\Lambda}n\tilde{\rho}_{ij}^{2}(k)\geq
x^{2}\right\\}\leq C|\Lambda|\\{1-\Phi(x)\\}+O(p^{-2}).$

  * (2)

if (C1)-(ii) holds, we have

$\displaystyle\operatorname{P}\left\\{\max_{(i,j,k)\in\Lambda}n\tilde{\rho}_{ij}^{2}(k)\geq
x^{2}\right\\}\leq C|\Lambda|\\{1-\Phi(x)\\}+{O(n^{-\epsilon/8}})$

uniformly for $0\leq x\leq\sqrt{8\log p}$ and $\Lambda\subset\\{(i,j,k):1\leq
i,j\leq p,1\leq k\leq K\\}$.

Proof. Rewrite

$\displaystyle n\tilde{\rho}_{ij}^{2}(k)=$
$\displaystyle\frac{1}{n}\left(\sum_{t=1}^{n-k}\sigma_{i}^{-1}\sigma_{j}^{-1}\varepsilon_{ti}\varepsilon_{t+k,j}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{\left(\sum_{t=1}^{n-k}\sigma_{i}^{-1}\sigma_{j}^{-1}\varepsilon_{ti}\varepsilon_{t+k,j}\right)^{2}}{\sum_{t=1}^{n-k}\sigma_{i}^{-2}\sigma_{j}^{-2}\varepsilon_{ti}^{2}\varepsilon_{t+k,j}^{2}}\times\frac{1}{n}\sum_{t=1}^{n-k}\sigma_{i}^{-2}\sigma_{j}^{-2}\varepsilon_{ti}^{2}\varepsilon_{t+k,j}^{2}.$

By the self-normalized large deviation theorem for independent random
variables (Theorem 1 in Jing et al. (2003)), we can get

$\displaystyle\max_{1\leq i\leq j\leq p,1\leq k\leq
K}\mathrm{P}\left\\{\frac{\left(\sum_{t=1}^{n-k}\sigma_{i}^{-1}\sigma_{j}^{-1}\varepsilon_{ti}\varepsilon_{t+k,j}\right)^{2}}{\sum_{t=1}^{n-k}\sigma_{i}^{-2}\sigma_{j}^{-2}\varepsilon_{ti}^{2}\varepsilon_{t+k,j}^{2}}\geq
x^{2}\right\\}\leq C\\{1-\Phi(x)\\}$ (16)

uniformly for $0\leq x\leq(8\log p)^{1/2}$. By Lemma 3 in Cai et al. (2013),
we have

  * (1)

if (C1)-(i) holds, we have

$\displaystyle\operatorname{P}\left(\left|\frac{1}{n}\sum_{t=1}^{n-k}\sigma_{i}^{-2}\sigma_{j}^{-2}\varepsilon_{ti}^{2}\varepsilon_{t+k,j}^{2}-1\right|\geq
C\frac{{\color[rgb]{0,0,0}\epsilon_{n}}}{\log p}\right)=O(p^{-\delta}),$ (17)

  * (2)

if (C1)-(ii) holds,

$\displaystyle\operatorname{P}\left(\left|\frac{1}{n}\sum_{t=1}^{n-k}\sigma_{i}^{-2}\sigma_{j}^{-2}\varepsilon_{ti}^{2}\varepsilon_{t+k,j}^{2}-1\right|\geq
C\frac{{\color[rgb]{0,0,0}\epsilon_{n}}}{\log
p}\right)=O(n^{-{\color[rgb]{0,0,0}\epsilon/8}}),$ (18)

for any $\delta>0$. Thus, together (16), (17) with (18), we can complete the
proof of this lemma. $\square$

Now, we are ready to present the proof of Theorem 1.

Proof of Theorem 1 Define $\tilde{T}_{n}\doteq\max_{1\leqslant k\leqslant
K}\tilde{T}_{n,k},$ where $\tilde{T}_{n,k}\doteq\max_{1\leqslant i,j\leqslant
p}n^{1/2}\left|\tilde{\rho}_{ij}(k)\right|$. Conditional on the event
$\\{\max_{1\leq i\leq p}\left|\hat{\sigma}_{i}^{2}-\sigma_{i}^{2}\right|\geq
C\frac{\epsilon_{n}}{\log p}\\}$, we have

$\displaystyle|T_{n}^{2}-\tilde{T}_{n}^{2}|\leq
C\tilde{T}_{n}\frac{\epsilon_{n}}{\log p}.$

Thus, by Lemma 3, we only need to show that

$\operatorname{P}\left\\{\tilde{T}_{n}-2\log(Kp^{2})+\log\log(Kp^{2})\leqslant
y\right\\}\to\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}.$

Restate that
$\tilde{\rho}_{ij}(k)=\frac{1}{n}\sum_{t=1}^{n-k}\sigma_{i}^{-1}\sigma_{j}^{-1}\varepsilon_{ti}\varepsilon_{t+k,j}$.
Without loss of generality, we assume that $\sigma_{i}=1$ for all $i$. After
some simply calculation, we have
$\mathrm{cov}\left\\{n^{1/2}\tilde{\rho}_{ij}(k),n^{1/2}\tilde{\rho}_{sw}(l)\right\\}=\rho_{is}\rho_{jw}\mathbb{I}(k=l)$.
Thus, we rearrange $\\{n^{1/2}\tilde{\rho}_{ij}(k)\\}_{1\leq i,j\leq p,1\leq
k\leq K}$ as $\\{\nu_{1},\cdots,\nu_{N}\\}$ with $N=Kp^{2}$. Let
$\bm{\nu}=(\nu_{1},\cdots,\nu_{N})^{\top}$, then

$\displaystyle\mathrm{cov}(\bm{\nu})=\\{a_{ij}\\}_{1\leq i,j\leq
N}\doteq\mathrm{diag}\\{{\color[rgb]{0,0,0}\bm{\Gamma}(0)\otimes\bm{\Gamma}(0),\cdots,\bm{\Gamma}(0)\otimes\bm{\Gamma}(0)}\\},$
(19)

where $\otimes$ denotes the Kronecker product and
$\mathrm{diag}\\{\bm{\Gamma}(0)\otimes\bm{\Gamma}(0),\cdots,\bm{\Gamma}(0)\otimes\bm{\Gamma}(0)\\}$
denotes the block diagonal matrix composed by
$\bm{\Gamma}(0)\otimes\bm{\Gamma}(0),\cdots,\bm{\Gamma}(0)\otimes\bm{\Gamma}(0)$.
For $1\leq i\leq N$, define $B_{N,i}=\big{\\{}1\leq j\leq
N:|a_{ij}|\geq\delta_{p}\big{\\}}$ and $C_{N}=\big{\\{}i:|B_{N,i}|\geq
p^{\kappa}\big{\\}}$. By the definition of $a_{ij}$, we have
$|C_{N}|=|C_{p}|$.

Define $z=\\{2\log(N)-2\log\log(N)+y\\}^{1/2}$. By Lemma 4, if (C1)-(i) holds,
we have

$\displaystyle\textrm{P}(|\nu_{i}|\geq z)\leq C\\{1-\Phi(z)\\}+O(p^{-2})\leq
C\frac{1}{\sqrt{\pi}}\frac{e^{-y/2}}{N}+O(p^{-2});$

if (C1)-(ii) holds, we have

$\displaystyle\textrm{P}(|\nu_{i}|\geq z)\leq
C\\{1-\Phi(z)\\}+O({\color[rgb]{0,0,0}n^{-\epsilon/8}})\leq
C\frac{1}{\sqrt{\pi}}\frac{e^{-y/2}}{N}+O(n^{-\epsilon/8}).$

Thus, by Condition (C3),

$\textrm{P}\left(\max_{i\in
C_{N}}\left|\nu_{i}\right|>z\right)\leq\left|C_{N}\right|\cdot\textrm{P}(|\nu_{i}|\geq
z)\rightarrow 0$

as $n$, $p\rightarrow\infty.$ Set $D_{N}=\left\\{1\leq i\leq
N:\left|B_{N,i}\right|<p^{\kappa}\right\\}.$ By Condition (C3),
$\left|D_{N}\right|/N\rightarrow 1$ as $N\rightarrow\infty$. Obviously,

$\displaystyle\textrm{P}\left(\max_{i\in D_{N}}\left|\nu_{i}\right|>z\right)$
$\displaystyle\leq\textrm{P}\left(\max_{1\leq i\leq
N}\left|\nu_{i}\right|>z\right)\leq\textrm{P}\left(\max_{i\in
D_{N}}\left|\nu_{i}\right|>z\right)+\textrm{P}\left(\max_{i\in
C_{N}}\left|\nu_{i}\right|>z\right).$

Therefore, to prove this theorem, it is enough to show

$\lim_{N\rightarrow\infty}\textrm{P}\left(\max_{i\in
D_{N}}\left|\nu_{i}\right|>z\right)=1-\exp\left(-\frac{1}{\sqrt{\pi}}e^{-x/2}\right)$

as $N\rightarrow\infty$.

We redefine
$\nu_{s}=n^{1/2}\tilde{\rho}_{ij}(k)=n^{-1/2}\sum_{t=1}^{n-k}\varepsilon_{ti}\varepsilon_{t+k,j}=n^{-1/2}\sum_{t=1}^{n-k}Z_{ts}$.
Let
$\breve{Z}_{ts}={Z}_{ts}\mathbb{I}({Z}_{ts}\leq\tau_{n})-\mathbb{E}\\{{Z}_{ts}I({Z}_{ts}\leq\tau_{n})\\}$.
Here, $\tau_{n}=\eta^{-1}8M\log(p+n)$, if (C1)-(i) holds, and
$\tau_{n}=\sqrt{n}/(\log p)^{8}$, if (C1)-(ii) holds. Define
$\breve{\nu}_{i}=n^{-1/2}\sum_{t=1}^{n-k}\breve{Z}_{ts}$ and $q=|D_{N}|$. If
(C1)-(i) holds, then

$\displaystyle\max_{1\leq k\leq
q}\frac{1}{\sqrt{n}}\sum_{l=1}^{n}\mathbb{E}\left|Z_{lk}\right|\mathbb{I}\left\\{\left|Z_{lk}\right|\geq\eta^{-1}8M\log(p+n)\right\\}$
$\displaystyle\leq$ $\displaystyle C\sqrt{n}\max_{1\leq l\leq n}\max_{1\leq
k\leq
q}\mathbb{E}\left|Z_{lk}\right|\mathbb{I}\left\\{\left|Z_{lk}\right|\geq\eta^{-1}8M\log(p+n)\right\\}$
$\displaystyle\leq$ $\displaystyle C\sqrt{n}(p+n)^{-4}\max_{1\leq l\leq
n}\max_{1\leq k\leq
q}\mathbb{E}\left|Z_{lk}\right|\exp\left\\{\eta\left|Z_{lk}\right|/\left(2M\right)\right\\}$
$\displaystyle\leq$ $\displaystyle C\sqrt{n}(p+n)^{-4}.$

If (C1)-(ii) holds, then

$\displaystyle\max_{1\leq k\leq
q}\frac{1}{\sqrt{n}}\sum_{l=1}^{n}\mathbb{E}\left|Z_{lk}\right|\mathbb{I}\left\\{\left|Z_{lk}\right|\geq\sqrt{n}/(\log
p)^{8}\right\\}$ $\displaystyle\leq$ $\displaystyle C\sqrt{n}\max_{1\leq l\leq
n}\max_{1\leq k\leq
q}\mathbb{E}\left|Z_{lk}\right|\mathbb{I}\left\\{\left|Z_{lk}\right|\geq\sqrt{n}/(\log
p)^{8}\right\\}\leq Cn^{-\gamma_{0}-\epsilon/8}.$

Thus, we have

$\displaystyle\operatorname{P}\left\\{\max_{1\leq k\leq
q}|\nu_{k}-\breve{\nu}_{k}|\geq(\log p)^{-1}\right\\}$ $\displaystyle\leq$
$\displaystyle\operatorname{P}\left(\max_{1\leq k\leq q}\max_{1\leq l\leq
n}|Z_{lk}|\geq\tau_{n}\right)$ $\displaystyle\leq$ $\displaystyle
n\operatorname{P}(\max_{1\leq i,j\leq p}\max_{1\leq s\leq
K}|\varepsilon_{ti}\varepsilon_{t+s,j}|\geq\tau_{n})$ $\displaystyle\leq$
$\displaystyle n\max_{1\leq i,j\leq p}\max_{1\leq s\leq
K}\left\\{\operatorname{P}(|\varepsilon_{ti}|\geq\tau_{n}^{1/2})+\operatorname{P}(|\varepsilon_{t+s,j}|\geq\tau_{n}^{1/2})\right\\}$
$\displaystyle=$ $\displaystyle O(p^{-1}+n^{-\epsilon/8}).$

Note that

$\displaystyle\left|\max_{1\leq k\leq q}\nu_{k}^{2}-\max_{1\leq k\leq
q}\breve{\nu}_{k}^{2}\right|\leq 2\max_{1\leq k\leq
q}|\breve{\nu}_{k}|\max_{1\leq k\leq q}|\nu_{k}-\breve{\nu}_{k}|+\max_{1\leq
k\leq q}\left|\nu_{k}-\breve{\nu}_{k}\right|^{2}.$ (20)

Therefore, to prove this theorem, it is enough to show

$\lim_{N\rightarrow\infty}\operatorname{P}\left(\max_{i\in
D_{N}}\left|\breve{\nu}_{i}\right|>z\right)=1-\exp\left(-\frac{1}{\sqrt{\pi}}e^{-x/2}\right)$

as $N\rightarrow\infty$. Then, by Bonferroni inequality,

$\sum_{t=1}^{2k}(-1)^{t-1}\alpha_{t}\leq\textrm{P}\left(\max_{i\in
D_{N}}\left|\breve{\nu}_{i}\right|>z\right)\leq\sum_{t=1}^{2k+1}(-1)^{t-1}\alpha_{t}$

for any $k\geq 1$, where

$\alpha_{t}\doteq\sum^{*}\textrm{P}\left(\left|\breve{\nu}_{i_{1}}\right|>z,\cdots,\left|\breve{\nu}_{i_{t}}\right|>z\right)$

for $1\leq t\leq N,$ and the sum runs over all $i_{1}<\cdots<i_{t}$ and
$i_{1},\cdots,i_{t}\in D_{N}.$ First, we will prove that

$\displaystyle\lim_{N\rightarrow\infty}\alpha_{t}=\frac{1}{t!}\pi^{-t/2}e^{-ty/2}$
(21)

for each $t\geq 1.$ All the assumptions in Theorem 1.1 in  Zaitsev (1987) are
satisfied. Thus, we have

$\displaystyle\sum^{*}\operatorname{P}\left\\{\left|Z_{i_{1}}\right|>z+\zeta_{n}(\log
N)^{-1/2},\cdots,\left|Z_{i_{t}}\right|>z+\zeta_{n}(\log N)^{-1/2}\right\\}$
$\displaystyle-\left(\begin{array}[]{c}|D_{N}|\\\
t\end{array}\right)c_{1}t^{5/2}\exp\left\\{-\frac{n^{1/2}\zeta_{n}}{c_{2}t^{3}(\log
N)^{1/2}}\right\\}$ $\displaystyle\leq$
$\displaystyle\sum^{*}\operatorname{P}\left\\{\left|\breve{\nu}_{i_{1}}\right|>z,\cdots,\left|\breve{\nu}_{i_{t}}\right|>z\right\\}$
$\displaystyle\leq$
$\displaystyle\sum^{*}\operatorname{P}\left\\{\left|Z_{i_{1}}\right|>z-\zeta_{n}(\log
N)^{-1/2},\cdots,\left|Z_{i_{t}}\right|>z-\zeta_{n}(\log N)^{-1/2}\right\\}$
$\displaystyle+\left(\begin{array}[]{c}|D_{N}|\\\
t\end{array}\right)c_{1}t^{5/2}\exp\left\\{-\frac{n^{1/2}\zeta_{n}}{c_{2}t^{3}(\log
N)^{1/2}}\right\\},$

where $(Z_{i_{1}},\cdots,Z_{i_{t}})^{\top}$ follows a multivariate normal
distribution with mean zero and the same covariance matrix with
$(\breve{\nu}_{i_{1}},\cdots,\breve{\nu}_{i_{t}})^{\top}$. By Lemma 1, we have

$\displaystyle\sum^{*}\operatorname{P}\left\\{\left|Z_{i_{1}}\right|>z+\zeta_{n}(\log
N)^{-1/2},\cdots,\left|Z_{i_{t}}\right|>z+\zeta_{n}(\log
N)^{-1/2}\right\\}\to\frac{1}{t!}\pi^{-t/2}e^{-ty/2},$
$\displaystyle\sum^{*}\operatorname{P}\left\\{\left|Z_{i_{1}}\right|>z-\zeta_{n}(\log
N)^{-1/2},\cdots,\left|Z_{i_{t}}\right|>z-\zeta_{n}(\log
N)^{-1/2}\right\\}\to\frac{1}{t!}\pi^{-t/2}e^{-ty/2},$

with $\zeta_{n}\to 0$ and $N\to\infty$. Additionally,

$\displaystyle\left(\begin{array}[]{c}|D_{N}|\\\
t\end{array}\right)c_{1}t^{5/2}\exp\left\\{-\frac{n^{1/2}\zeta_{n}}{c_{2}t^{3}(\log
N)^{1/2}}\right\\}$ $\displaystyle\leq$ $\displaystyle
C\left(\begin{array}[]{c}N\\\
t\end{array}\right)t^{5/2}\exp\left\\{-\frac{n^{1/2}\zeta_{n}}{c_{2}t^{3}(\log
N)^{1/2}}\right\\}\to 0$

for $\zeta_{n}\to 0$ sufficiently slow. Thus, we have

$\displaystyle\sum^{*}\operatorname{P}\left(\left|\breve{\nu}_{i_{1}}\right|>z,\cdots,\left|\breve{\nu}_{i_{t}}\right|>z\right)\to\frac{1}{t!}\pi^{-t/2}e^{-ty/2}.$

Let $N\rightarrow\infty$, we have

$\displaystyle\sum_{t=1}^{2k}(-1)^{t-1}\frac{1}{t!}\left(\frac{1}{\sqrt{\pi}}e^{-x/2}\right)^{t}$
$\displaystyle\leq$
$\displaystyle\liminf_{N\rightarrow\infty}\operatorname{P}\left(\max_{i\in
D_{N}}\left|\breve{\nu}_{i}\right|>z\right)$ $\displaystyle\leq$
$\displaystyle\limsup_{N\rightarrow\infty}\operatorname{P}\left(\max_{i\in
D_{N}}\left|\breve{\nu}_{i}\right|>z\right)\leq\sum_{t=1}^{2k+1}(-1)^{t-1}\frac{1}{t!}\left(\frac{1}{\sqrt{\pi}}e^{-x/2}\right)^{t}$

for each $k\geq 1.$ By letting $k\rightarrow\infty$ and using the Taylor
expansion of the function $1-e^{-x}$, we obtain the result. $\square$

###  6.2 Proof of Theorem 2

Define $(i_{0},j_{0},k_{0})=\arg\max_{1\leq i<j\leq p,1\leq k\leq
K}|\rho_{ij}(k)|$. Let
$\gamma_{il}=\mathbb{E}(\varepsilon_{it}^{2}\varepsilon_{i,t+l}^{2})-\sigma_{i}^{4}$.
By the condition of Theorem 2, the long-run variance
$\gamma_{i}^{L}=\lim_{n\to\infty}\\{\gamma_{i0}+2\sum_{l=1}^{n}(1-l/n)\gamma_{il}\\}$
is bounded. Hence,

$\displaystyle\mathbb{E}\left(\frac{1}{n}\sum_{t=1}^{n-k_{0}}\varepsilon_{i_{0}t}^{2}\right)=$
$\displaystyle\sigma_{i_{0}}^{2},$
$\displaystyle\mathrm{var}\left(\frac{1}{n}\sum_{t=1}^{n-k_{0}}\varepsilon_{i_{0}t}^{2}\right)=$
$\displaystyle\frac{1}{n}\\{\mathbb{E}(\varepsilon_{i_{0}t}^{4})-\sigma_{i_{0}}^{4}\\}+\frac{2}{n^{2}}\sum_{s<t}\\{\mathbb{E}(\varepsilon_{i_{0}t}^{2}\varepsilon_{i_{0}s}^{2})-\sigma_{i_{0}}^{4}\\}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\gamma_{i_{0}}+\frac{2}{n}\sum_{l=1}^{n-1}(1-l/n)\gamma_{i_{0}l}\to
0.$

Thus, we have
$\frac{1}{n}\sum_{t=1}^{n-k_{0}}\varepsilon_{i_{0}t}^{2}\mathop{\rightarrow}\limits^{p}\sigma_{i_{0}}^{2}$.
Similarly, we have

$\frac{1}{n}\sum_{t=1}^{n-k_{0}}\varepsilon_{j_{0}t}^{2}\mathop{\rightarrow}\limits^{p}\sigma_{j_{0}}^{2}$
and
$\frac{1}{n}\sum_{t=1}^{n-k_{0}}\varepsilon_{i_{0}t}\varepsilon_{j_{0},t+k_{0}}\mathop{\rightarrow}\limits^{p}\sigma_{i_{0}j_{0}}(k_{0})$.

Thus,
$\hat{\rho}_{i_{0}j_{0}}(k_{0})\mathop{\rightarrow}\limits^{p}\rho_{i_{0}j_{0}}(k_{0})$.
As $n,p\to\infty$, we have

$\displaystyle\operatorname{P}\left\\{\max_{1\leq k\leq K}\max_{1\leq i<j\leq
p}n\hat{\rho}^{2}_{ij}(k)-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right\\}$ $\displaystyle\geq$
$\displaystyle\operatorname{P}\left\\{n\hat{\rho}^{2}_{i_{0}j_{0}}(k_{0})-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right\\}$ $\displaystyle\to$
$\displaystyle\operatorname{P}\left\\{n{\rho}^{2}_{i_{0}j_{0}}(k_{0})-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right\\}=1$

by the condition ${\rho}_{i_{0}j_{0}}(k_{0})\geq 3\sqrt{\log
p/n}>\sqrt{2\log(Kp^{2})/n}$. $\square$

###  6.3 Proof of Theorem 3

Define $\bm{X}=(\bm{\varepsilon}_{\cdot
1}^{\top},\cdots,\bm{\varepsilon}_{\cdot p}^{\top})^{\top}\in\mathbb{R}^{d}$,
$d=np$ and $\bm{\varepsilon}_{\cdot
i}=(\varepsilon_{1i},\cdots,\varepsilon_{ni})^{\top}$. Consider the Gaussian
setting and a simple alternative set of parameters

$\mathcal{F}(\rho)\doteq\left\\{{\bf\Xi}:{\bf\Xi}=\mathrm{diag}\\{\underbrace{{\bf
I}_{n},\cdots,{\bf I}_{n}}_{k-1},{\bm{\Sigma}}(\rho),{\bf I}_{n},\cdots,{\bf
I}_{n}\\},1\leq k\leq p\right\\},$

where

$\displaystyle{\bm{\Sigma}}(\rho)=\left(\begin{array}[]{llllll}1&\rho&0&\cdots&0&0\\\
\rho&1&\rho&\cdots&0&0\\\ 0&\rho&1&\cdots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\ 0&0&0&\cdots&1&\rho\\\
0&0&0&\cdots&\rho&1\end{array}\right).$

Let $\mu_{\rho}$ be the uniform measure on $\mathcal{F}(\rho)$ and
$\rho=c_{0}(\log d/n)^{1/2}$ for some small enough constant $c_{0}<$ 1\. Let
$\mathrm{pr}_{{\bf\Xi}}$ denote the probability measure of
$N_{d}(\bm{0},{\bf\Xi})$ and
$\mathrm{pr}_{\mu_{\rho}}=\int\mathrm{pr}_{{\bf\Xi}}\mathrm{d}\mu_{\rho}({\bf\Xi}).$
Let $\mathrm{pr}_{0}$ denote the probability measure of $N_{d}\left(0,{\bf
I}_{d}\right).$ Note that, for any set $A,$ we have

$\sup_{{\bf\Xi}\in\mathcal{F}(\rho)}\operatorname{pr}_{{\bf\Xi}}\left(A^{C}\right)\geq\operatorname{pr}_{\mu_{\rho}}\left(A^{C}\right),~{}~{}1=\operatorname{pr}_{\mu_{\rho}}\left(A^{C}\right)+\operatorname{pr}_{\mu_{\rho}}(A)$

and

$\operatorname{pr}_{\mu_{\rho}}(A)\leq\operatorname{pr}_{0}(A)+\left|\operatorname{pr}_{\mu_{\rho}}(A)-\operatorname{pr}_{0}(A)\right|.$

Letting $A=\left\\{T_{\alpha}=1\right\\},$ the above equations yield

$\displaystyle\inf_{T_{\alpha}\in\mathcal{T}_{\alpha},{\bf\Xi}\in\mathcal{F}(\rho)}\sup_{\operatorname{pr}_{{\bf\Xi}}}\left(T_{\alpha}=0\right)\geq$
$\displaystyle
1-\alpha-\sup_{A:\mathrm{pr}_{0}(A)\leq\alpha}\left|\operatorname{pr}_{\mu_{\rho}}(A)-\operatorname{pr}_{0}(A)\right|$
$\displaystyle\geq$ $\displaystyle
1-\alpha-\frac{1}{2}\left\|\mathrm{pr}_{\mu_{\rho}}-\mathrm{pr}_{0}\right\|_{TV},$

where $\|\cdot\|_{TV}$ denotes the total variation norm. Setting
$L_{\mu_{\rho}}(y)=\operatorname{dpr}_{\mu_{\rho}}(y)/\mathrm{d}\mathrm{pr}_{0}(y),$
and by Jensen’s inequality, we have

$\displaystyle\left\|\mathrm{pr}_{\mu_{\rho}}-\mathrm{pr}_{0}\right\|_{TV}=$
$\displaystyle\int\left|L_{\mu_{\rho}}(y)-1\right|\mathrm{d}\mathrm{pr}_{0}(y)$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\mathrm{pr}_{0}}\left|L_{\mu_{\rho}}(Y)-1\right|\leq\left[\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}-1\right]^{1/2}.$

Therefore, as long as
$\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}=1+o(1),$
we have

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha},{\bf\Xi}\in\mathcal{F}(\rho)}\sup_{\mathrm{pr}_{{\bf\Xi}}}\left(T_{\alpha}=0\right)\geq
1-\alpha-o(1)>0.$

We then prove that
$\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}=1+o(1).$
By construction, we have

$L_{\mu_{\rho}}=\frac{1}{p}\sum_{{\bf\Xi}\in\mathcal{F}(\rho)}\left[\frac{1}{|{\bf\Xi}|^{1/2}}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left({\bf\Omega}-{\bf
I}_{d}\right){\color[rgb]{0,0,0}Z_{i,\cdot}}\right\\}\right],$

where ${\bf\Omega}={\bf\Xi}^{-1}$ and  $\left\\{Z_{i,\cdot}:1\leq i\leq
n\right\\}$ are independent and identically distributed as
$\mathcal{N}_{d}\left(0,{\bf I}_{d}\right)$. We have

$\displaystyle\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\sum_{{\bf\Xi}_{1},{\bf\Xi}_{2}\in\mathcal{F}(\rho)}\mathbb{E}\left[\frac{1}{\left|{\bf\Xi}_{1}\right|^{1/2}}\frac{1}{\left|{\bf\Xi}_{2}\right|^{1/2}}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left({\bf\Omega}_{1}+{\bf\Omega}_{2}-2I_{d}\right)Z_{i,\cdot}\right\\}\right],$

where ${\bf\Omega}_{i}={\bf\Xi}_{i}^{-1}$ for $i=1,2$. We write

$\displaystyle\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}=$
$\displaystyle\frac{p-1}{p}\mathbb{E}\left[\frac{1}{\left|{\bf\Xi}_{1}\right|^{1/2}}\frac{1}{\left|{\bf\Xi}_{2}\right|^{1/2}}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left({\bf\Omega}_{1}+{\bf\Omega}_{2}-2{\bf
I}_{d}\right)Z_{i,\cdot}\right\\}\right]$
$\displaystyle+\frac{1}{p}\mathbb{E}\left[\frac{1}{\left|{\bf\Xi}\right|}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left(2{\bf\Omega}-2{\bf
I}_{d}\right)Z_{i,\cdot}\right\\}\right]$ $\displaystyle\doteq$ $\displaystyle
E_{1}+E_{2},$

where $E_{1}$ represents the set of $\left({\bf\Xi}_{1},{\bf\Xi}_{2}\right)$
with ${\bf\Xi}_{1}\neq{\bf\Xi}_{2},$ and $E_{2}$ represents the set of
$\left({\bf\Xi}_{1},{\bf\Xi}_{2}\right)$ with ${\bf\Xi}_{1}={\bf\Xi}_{2}$. By
standard argument in moment generating functions of the Gaussian quadratic
form, we have

$\displaystyle\mathbb{E}\left\\{\exp\left(-\frac{1}{2}{\bm{W}}^{\top}{\bf
A}{\bm{W}}\right)\right\\}=$
$\displaystyle{\color[rgb]{0,0,0}\left[\\{1+\lambda_{1}({\bf
A})\\}\cdots\\{1+\lambda_{q}({\bf A})\\}\right]^{-1/2}}$ $\displaystyle=$
$\displaystyle\\{\det({\bf I}_{q}+{\bf A})\\}^{-1/2}$

if ${\bm{W}}\sim\mathcal{N}(\bm{0},{\bf I}_{q})$ and ${\bf
A}\in\mathbb{R}^{q\times q}$. Without loss of generality, define
${\bf\Xi}_{1}=\mathrm{diag}\\{{\bm{\Sigma}}(\rho),{\bf I}_{n},\cdots,{\bf
I}_{n}\\}$ and ${\bf\Xi}_{2}=\mathrm{diag}\\{{\bf
I}_{n},{\bm{\Sigma}}(\rho),{\bf I}_{n},\cdots,{\bf I}_{n}\\}$. Thus,
$|{\bf\Xi}_{1}|=|{\bf\Xi}_{2}|=|{\bm{\Sigma}}(\rho)|$. Additionally, define
${\bf\Omega}_{1}=\mathrm{diag}\\{{\bm{\Sigma}}(\rho)^{-1},{\bf
I}_{n},\cdots,{\bf I}_{n}\\}$ and ${\bf\Omega}_{2}=\mathrm{diag}\\{{\bf
I}_{n},{\bm{\Sigma}}(\rho)^{-1},{\bf I}_{n},\cdots,{\bf I}_{n}\\}$.

$\displaystyle\mathbb{E}\left[\frac{1}{\left|{\bf\Xi}_{1}\right|^{1/2}}\frac{1}{\left|{\bf\Xi}_{2}\right|^{1/2}}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left({\bf\Omega}_{1}+{\bf\Omega}_{2}-2I_{d}\right)Z_{i,\cdot}\right\\}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\frac{1}{\left|{\bf\Xi}_{1}\right|^{1/2}}\frac{1}{\left|{\bf\Xi}_{2}\right|^{1/2}}\exp\left[-\frac{1}{2}Z_{1,\cdot}^{\mathrm{T}}\left\\{{\bm{\Sigma}}(\rho)^{-1}-{\bf
I}_{n}\right\\}Z_{1,\cdot}-\frac{1}{2}Z_{2,\cdot}^{\mathrm{T}}\left\\{{\bm{\Sigma}}(\rho)^{-1}-{\bf
I}_{n}\right\\}Z_{2,\cdot}\right]\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left|{\bm{\Sigma}}(\rho)\right|}|{\bm{\Sigma}}(\rho)^{-1}-{\bf
I}_{n}+{\bf I}_{n}|^{-1/2}|{\bm{\Sigma}}(\rho)^{-1}-{\bf I}_{n}+{\bf
I}_{n}|^{-1/2}=1.$

Thus,

$\displaystyle E_{1}=\frac{p-1}{p}=1+o(1)$ (22)

as $p\to\infty$. Similarly,

$\displaystyle\frac{1}{p}\mathbb{E}\left[\frac{1}{\left|{\bf\Xi}\right|}\exp\left\\{-\frac{1}{2}Z_{i,\cdot}^{\mathrm{T}}\left(2{\bf\Omega}-2I_{d}\right)Z_{i,\cdot}\right\\}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{p}\frac{1}{|{\bm{\Sigma}}(\rho)|}\mathbb{E}\left[\exp\left\\{-\frac{1}{2}Z_{1,\cdot}^{\mathrm{T}}\left(2{\bm{\Sigma}}(\rho)^{-1}-2{\bf
I}_{n}\right)Z_{1,\cdot}\right\\}\right]$ $\displaystyle=$
$\displaystyle\frac{1}{p}\frac{1}{|{\bm{\Sigma}}(\rho)|}|2{\bm{\Sigma}}(\rho)^{-1}-{\bf
I}_{n}|^{-1/2}.$

By (10.1) in  Daniels (1956), we have
$|{\bm{\Sigma}}(\rho)|=(1-\rho^{2})^{n-1}$. Define

$\displaystyle{\bf A}=\left(\begin{array}[]{llllll}0&1&0&\cdots&0&0\\\
1&0&1&\cdots&0&0\\\ 0&1&0&\cdots&0&0\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\cdots&0&1\\\ 0&0&0&\cdots&1&0\end{array}\right).$

Then, ${\bm{\Sigma}}(\rho)={\bf I}_{n}+\rho{\bf A}$. By the Taylor expansion,
we have $({\bf I}_{n}+\rho{\bf A})^{-1}=\sum_{k=0}^{\infty}(-\rho)^{k}{\bf
A}^{k}$ and

$\displaystyle 2{\bm{\Sigma}}(\rho)^{-1}-{\bf I}_{n}=$ $\displaystyle 2({\bf
I}_{n}+\rho{\bf A})^{-1}-{\bf I}_{n}=2({\bf I}_{n}-\rho{\bf A}+\rho^{2}{\bf
A}^{2}-\rho^{3}{\bf A}^{3}+\cdots)-{\bf I}_{n}$ $\displaystyle=$
$\displaystyle{\bf I}_{n}-2\rho{\bf A}+2\rho^{2}{\bf
A}^{2}\sum_{k=0}^{\infty}(-\rho)^{k}{\bf A}^{k}$ $\displaystyle=$
$\displaystyle{\bf I}_{n}-2\rho{\bf A}+2\rho^{2}{\bf A}^{2}({\bf
I}_{n}+\rho{\bf A})^{-1}.$

Because $\rho^{2}{\bf A}^{2}({\bf I}_{n}+\rho{\bf A})^{-1}$ is positive
definite, we have

$\displaystyle|2{\bm{\Sigma}}(\rho)^{-1}-{\bf I}_{n}|\geq|{\bf
I}_{n}-2\rho{\bf A}|=(1-4\rho^{2})^{n-1}.$

Thus,

$\displaystyle E_{2}\leq
p^{-1}(1-\rho^{2})^{-n+1}(1-4\rho^{2})^{-(n-1)/2}=p^{-1}\exp(3c_{0}^{2}\log
p)\\{1+o(1)\\}\to 0$ (23)

if $\rho=c_{0}(\log p/n)^{1/2}$ and $c_{0}^{2}<1/3$. Combining (22) and (23),
we have
$\mathbb{E}_{\mathrm{pr}_{0}}\left\\{L_{\mu_{\rho}}^{2}(Y)\right\\}=1+o(1).$
Lastly, we can easily show that for $\rho=c_{0}(\log p/n)^{1/2}$,

$\left[F(\bm{\varepsilon}):\mathrm{cor}_{F}\\{(\bm{\varepsilon}_{\cdot
1}^{\top},\cdots,\bm{\varepsilon}_{\cdot
p}^{\top})^{\top}\\}\in\mathcal{F}(\rho),F(\bm{\varepsilon})\mbox{ is
Gaussian}\right]\subset\left[F(\bm{\varepsilon}):R\\{F(\bm{\varepsilon})\\}\in\mathcal{U}(c)\right],$

where
$\bm{\varepsilon}=\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{n}\\}$ and
$R\\{F(\bm{\varepsilon})\\}=\big{\\{}\mathrm{cor}_{F}(\bm{\varepsilon}_{t+1},\bm{\varepsilon}_{t}),\cdots,\mathrm{cor}_{F}(\bm{\varepsilon}_{t+K},\bm{\varepsilon}_{t})\big{\\}}$.
Thus,

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{R\\{F(\bm{\varepsilon})\\}\in\mathcal{U}(c)}\operatorname{pr}_{{\bf\Xi}}\left(T_{\alpha}=0\right)\geq\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{{\bf\Xi}\in\mathcal{F}(\rho)}\operatorname{pr}_{{\bf\Xi}}\left(T_{\alpha}=0\right)\geq
1-\alpha-o(1)>0.$

This completes the proof. $\square$

###  6.4 Proof of Theorem 4

Firstly, we restate Lemma 2.1 in Srivastava (2009) on the quadratic forms.

######  Lemma 5.

Under Condition (C4), for any $m\times m$ symmetric matrix
$A=\\{a_{ij}\\}_{1\leq i,j\leq m}$ and $B=\\{b_{ij}\\}_{1\leq i,j\leq m}$ of
constants, we have

$\displaystyle\mathbb{E}\\{({\bm{z}}_{t}^{T}A{\bm{z}}_{t})^{2}\\}=$
$\displaystyle\Delta\sum_{i=1}^{m}a_{ii}^{2}+2\mathrm{tr}(A^{2})+\\{\mathrm{tr}(A)\\}^{2},$
$\displaystyle\mathrm{var}({\bm{z}}_{t}^{T}A{\bm{z}}_{t})=$
$\displaystyle\Delta\sum_{i=1}^{p}a_{ii}^{2}+2\mathrm{tr}(A^{2}),$
$\displaystyle\mathbb{E}\\{({\bm{z}}_{t}^{T}A{\bm{z}}_{t})({\bm{z}}_{t}^{T}B{\bm{z}}_{t})\\}=$
$\displaystyle\Delta\sum_{i=1}^{m}a_{ii}b_{ii}+2\mathrm{tr}(AB)+\mathrm{tr}(A)\mathrm{tr}(B),$

where $\Delta=\mathbb{E}(z_{it}^{4})-3$.

Now, we are ready to present the proof of Theorem 4.

Proof. Let

$\displaystyle
T_{\operatorname{SUM}}=\sum_{l=1}^{K}\frac{2}{n(n-1)}\underset{t<s}{\sum\sum}\bm{\varepsilon}_{t}^{\top}\bm{\varepsilon}_{s}\bm{\varepsilon}_{t+l}^{\top}\bm{\varepsilon}_{s+l}\doteq\sum_{l=1}^{K}T_{l}.$

We will show that for each $l\in\\{1,\cdots,K\\}$,

$\displaystyle\frac{T_{l}}{\sqrt{\frac{2}{n(n-1)}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1).$
(24)

Define
$V_{nj}=n^{-1}(n-1)^{-1}\sum_{i=l+1}^{j-1}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l}\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{j}$,
$j\in\\{l+2,\cdots,n\\}$ and $W_{nk}=\sum_{i=l+2}^{k}V_{ni}$,
$k\in\\{l+2,\cdots,n\\}$. Let
$\mathcal{F}_{i}\doteq\sigma\\{\bm{\varepsilon}_{1},\cdots,\bm{\varepsilon}_{i}\\}$
be the $\sigma$-field generated by $\\{\bm{\varepsilon}_{j}\\}_{j\leq i}$. It
is easy to show that $\mathbb{E}(V_{ni}|\mathcal{F}_{i-1})=0$ and it follows
that $\\{W_{nk},\mathcal{F}_{k}:l+2\leq k\leq n\\}$ is a zero mean martingale.
Let $v_{ni}=\mathbb{E}(V_{ni}^{2}|\mathcal{F}_{i-1})$, $l+2\leq i\leq n$ and
$V_{n}=\sum_{i=l+2}^{n}v_{ni}$. The central limit theorem (Hall and Hyde,
1980) will hold if we can show

$\displaystyle\frac{V_{n}}{\mathrm{var}(W_{nn})}\mathop{\rightarrow}\limits^{p}1,$
(25)

and for any $\epsilon>0$,

$\displaystyle\sum_{i=l+2}^{n}n^{2}\mathrm{tr}^{-2}({\bm{\Sigma}}^{2})\mathbb{E}\left[V_{ni}^{2}\mathbb{I}\left\\{|V_{ni}|>\epsilon\sqrt{n^{-2}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}\right\\}|\mathcal{F}_{i-1}\right]\mathop{\rightarrow}\limits^{p}0.$
(26)

It can be shown that

$\displaystyle v_{ni}$ $\displaystyle=$
$\displaystyle\frac{1}{n^{2}(n-1)^{2}}\left\\{\sum_{j=l+1}^{i-1}(\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l})^{2}\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{j}+2\sum_{l+1\leq
j<k<i}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{k-l}\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{k}\right\\}.$

Then,

$\displaystyle\frac{V_{n}}{\mathrm{var}(W_{nn})}$ $\displaystyle=$
$\displaystyle\frac{2}{n(n-1)\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}\Big{\\{}\sum_{i=l+2}^{n}\sum_{j=l+1}^{i-1}(\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l})^{2}\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{j}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\sum_{i=l+2}^{n}\sum_{l+1\leq
j<k\leq
i}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{k-l}\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{k}\Big{\\}}$
$\displaystyle\doteq$ $\displaystyle C_{n1}+C_{n2}.$

Simple algebras lead to

$\displaystyle\mathbb{E}(C_{n1})$ $\displaystyle=\frac{(n-l)(n-l-1)}{n(n-1)},$
$\displaystyle\mathrm{var}(C_{n1})$
$\displaystyle=\frac{4}{n^{2}(n-1)^{2}\mathrm{tr}^{4}({\bm{\Sigma}}^{2})}\mathbb{E}\left[\sum_{i=l+2}^{n}\sum_{j=l+1}^{n-1}\left\\{(\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l})^{4}(\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{j})^{2}-\mathrm{tr}^{4}({\bm{\Sigma}}^{2})\right\\}\right].$

By Lemma 5, we have
$\mathbb{E}\left\\{(\bm{\varepsilon}_{j}^{T}{\bm{\Sigma}}\bm{\varepsilon}_{j})^{2}-\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\right\\}=O\\{\mathrm{tr}({\bm{\Sigma}}^{4})\\}$.
Next, we will show that
$\mathbb{E}\left\\{(\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l})^{4}-\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\right\\}=O\\{\mathrm{tr}({\bm{\Sigma}}^{4})\\}$.
Define
${\bm{\Sigma}}^{1/2}{\bm{\Sigma}}{\bm{\Sigma}}^{1/2}\doteq\\{\omega_{kl}\\}_{1\leq
k,l\leq p}$.

$\displaystyle\mathbb{E}\\{(\bm{\varepsilon}_{i}^{\top}\bm{\varepsilon}_{s})^{4}\\}=\mathbb{E}\left\\{({\bm{z}}_{i}^{T}{\bm{\Sigma}}{\bm{z}}_{s})^{4}\right\\}=\mathbb{E}\left\\{\left(\sum_{k,l=1}^{m}\sigma_{kl}z_{ik}z_{jl}\right)^{4}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{k,l=1}^{m}\sigma_{kl}^{4}\mathbb{E}(z_{ik}^{4})\mathbb{E}(z_{jl}^{4})+\sum_{k\not=l}^{m}\sum_{s\not=t}^{m}\sigma_{kl}^{2}\sigma_{st}^{2}\mathbb{E}(z_{ik}^{2})\mathbb{E}(z_{is}^{2})\mathbb{E}(z_{jl}^{2})\mathbb{E}(z_{jt}^{2})$
$\displaystyle+2\sum_{k=1}^{m}\sum_{s\not=t}^{m}\sigma_{ks}^{2}\sigma_{kt}^{2}\mathbb{E}(z_{ik}^{4})\mathbb{E}(z_{js}^{2}z_{jt}^{2})+\sum_{k\not=l}^{m}\sum_{s\not=t}^{m}\sigma_{kl}\sigma_{kt}\sigma_{st}\sigma_{sl}\mathbb{E}(z_{ik}^{2})\mathbb{E}(z_{jl}^{2})\mathbb{E}(z_{is}^{2})\mathbb{E}(z_{jt}^{2}).$

Note that
$\mathrm{tr}^{2}({\bm{\Sigma}}^{2})=(\sum_{s,t}\sigma_{st}^{2})^{2}=\sum_{k,l,s,t}\sigma_{st}^{2}\sigma_{kl}^{2}$
and

$\displaystyle\sum_{k,l=1}^{m}\sigma_{kl}^{4}$
$\displaystyle\leq\left(\sum_{k,l}\sigma_{kl}^{2}\right)^{2},$
$\displaystyle\sum_{k=1}^{m}\sum_{s\not=t}^{m}\sigma_{ks}^{2}\sigma_{kt}^{2}$
$\displaystyle\leq\left(\sum_{k,l}\sigma_{kl}^{2}\right)^{2},$
$\displaystyle\sum_{k\not=l}^{m}\sum_{s\not=t}^{m}\sigma_{kl}^{2}\sigma_{st}^{2}$
$\displaystyle\leq\sum_{k,l,s,t}\sigma_{st}^{2}\sigma_{kl}^{2},$
$\displaystyle\sum_{k\not=l}^{m}\sum_{s\not=t}^{m}\sigma_{kl}\sigma_{kt}\sigma_{st}\sigma_{sl}$
$\displaystyle\leq\sum_{k\not=l}\omega_{kl}^{2}\leq\sum_{k,l}\omega_{kl}^{2}=\mathrm{tr}({\bm{\Sigma}}^{4}).$

Thus, we have

$\displaystyle\mathbb{E}\\{(\bm{\varepsilon}_{i}^{\top}\bm{\varepsilon}_{s})^{4}\\}-\mathrm{tr}^{2}({\bm{\Sigma}}^{2})=O\\{\mathrm{tr}({\bm{\Sigma}}^{4})\\}.$
(27)

Hence, $\mathrm{var}(C_{n1})\to 0$ due to
$\mathrm{tr}({\bm{\Sigma}}^{4})=o\\{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\\}$.
Then, $C_{n1}\mathop{\rightarrow}\limits^{p}1$. Similarly,
$\mathbb{E}(C_{n2})=0$ and

$\displaystyle\mathrm{var}(C_{n2})=O(n^{-2})\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{4})}{\mathrm{tr}^{4}({\bm{\Sigma}}^{2})}\to
0,$

which implies $C_{n2}\mathop{\rightarrow}\limits^{p}0$. Thus, (25) holds.

It remains to show (26). Since

$\displaystyle\mathbb{E}\left[V_{ni}^{2}\mathbb{I}\left\\{|V_{ni}|>\epsilon\sqrt{n^{-2}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}\right\\}|\mathcal{F}_{i-1}\right]\leq\mathbb{E}(V_{ni}^{4}|\mathcal{F}_{i-1})/\\{\epsilon^{2}n^{-2}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\\},$

we only need to show that

$\displaystyle\sum_{i=l+2}^{n}\mathbb{E}(V_{ni}^{4})=o\\{n^{-4}\mathrm{tr}^{4}({\bm{\Sigma}}^{2})\\}.$

Note that

$\displaystyle\sum_{i=l+2}^{n}\mathbb{E}(V_{ni}^{4})=O(n^{-4})\sum_{i=l+2}^{n}\mathbb{E}\left\\{\left(\sum_{j=l+1}^{i-1}\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l}\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{j}\right)^{4}\right\\},$

which can be decomposed as $3Q+P$ with

$\displaystyle Q$
$\displaystyle=O(n^{-8})\sum_{i=l+2}^{n}\sum_{s\not=t}^{i-1}\mathbb{E}\left(\bm{\varepsilon}_{i-l}^{T}\bm{\varepsilon}_{s-l}\bm{\varepsilon}_{s-l}^{T}\bm{\varepsilon}_{i-l}\bm{\varepsilon}_{i-l}^{T}\bm{\varepsilon}_{t-l}\bm{\varepsilon}_{t-l}^{T}\bm{\varepsilon}_{i-l}\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{s}\bm{\varepsilon}_{s}^{T}\bm{\varepsilon}_{i}\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{t}\bm{\varepsilon}_{t}^{T}\bm{\varepsilon}_{i}\right),$
$\displaystyle P$
$\displaystyle=O(n^{-8})\sum_{i=l+2}^{n}\sum_{s=1}^{i-1}\mathbb{E}\left\\{(\bm{\varepsilon}_{i-l}^{\top}\bm{\varepsilon}_{j-l})^{4}(\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{s})^{4}\right\\}.$

Note that
$Q={O}(n^{-4})\mathbb{E}^{2}\\{(\bm{\varepsilon}_{i}^{T}\bm{\varepsilon}_{i})^{2}\\}=o\\{n^{-4}\mathrm{tr}^{4}({\bm{\Sigma}}^{2})\\}$
by Lemma 5. By (27), we have
$P=O\\{n^{-4}\mathrm{tr}^{2}({\bm{\Sigma}}^{4})\\}=o\\{n^{-4}\mathrm{tr}^{4}({\bm{\Sigma}}^{2})\\}$.
And then (26) follows immediately. This completes the proof of (24). Finally,
after some simple algebras, we have $\mathbb{E}(T_{l}T_{k})=0$ if $l\not=k$.
Thus, we have

$\displaystyle\frac{\sum_{l=1}^{K}T_{l}}{\sqrt{\frac{2K}{n(n-1)}\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1).$
(28)

Here, we complete the proof. $\square$

###  6.5 Proof of Proposition 1

Proof. Under $H_{0}$, due to Proposition A.2 in Chen et al. (2010) and the
condition
$\mathrm{tr}({\bm{\Sigma}}^{4})=o(\mathrm{tr}^{2}({\bm{\Sigma}}^{2}))$, we
have

$\displaystyle\mathbb{E}\\{\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}\\}=$
$\displaystyle\mathrm{tr}({\bm{\Sigma}}^{2}),$
$\displaystyle\mathrm{var}\\{\widehat{\mathrm{tr}({\bm{\Sigma}}^{2})}\\}=$
$\displaystyle
4n^{-2}\operatorname{tr}^{2}\left({\bm{\Sigma}}^{2}\right)+8n^{-1}\operatorname{tr}\left({\bm{\Sigma}}^{4}\right)+4\Delta
n^{-1}\operatorname{tr}\left({\color[rgb]{0,0,0}{\bm{\Sigma}}^{2}\circ{\bm{\Sigma}}^{2}}\right)$
$\displaystyle+O\left\\{n^{-3}\operatorname{tr}^{2}\left({\bm{\Sigma}}^{2}\right)+n^{-2}\operatorname{tr}\left({\bm{\Sigma}}^{4}\right)\right\\}=o\\{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})\\},$

where ${\bf A}\circ{\bf B}=\\{a_{ij}b_{ij}\\}$ for two matrix ${\bf
A}=\\{a_{ij}\\}$ and ${\bf B}=\\{b_{ij}\\}$ . Hence, we complete the proof of
this proposition. $\square$

###  6.6 Proof of Theorem 5

Proof. Recall that $\bm{\varepsilon}_{t}={\bf A}_{0}{\bm{z}}_{t}+{\bf
A}_{1}{\bm{z}}_{t-1}$ and $K=1$. Actually,

$\displaystyle G_{1}=$
$\displaystyle\frac{1}{n(n-1)}\sum_{s\not=t}^{T}\left({\bf
A}_{0}{\bm{z}}_{s}+{\bf A}_{1}{\bm{z}}_{s-1}\right)^{\top}\left({\bf
A}_{0}{\bm{z}}_{t}+{\bf A}_{1}{\bm{z}}_{t-1}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left({\bf
A}_{0}{\bm{z}}_{t-1}+{\bf A}_{1}{\bm{z}}_{t-2}\right)^{\top}\left({\bf
A}_{0}{\bm{z}}_{s-1}+{\bf A}_{1}{\bm{z}}_{s-2}\right)$ $\displaystyle=$
$\displaystyle G(I)+G(II)+G(III),$

where

$\displaystyle G(I)$ $\displaystyle\doteq$
$\displaystyle\frac{1}{n(n-1)}\sum_{s\not=t}^{T}\left({\bm{z}}_{s}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-1}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{0}{\bm{z}}_{s-1}+{\bm{z}}_{s-1}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-2}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{1}{\bm{z}}_{s-2}\right.$
$\displaystyle\left.+{\bm{z}}_{s}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-2}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{1}{\bm{z}}_{s-2}+{\bm{z}}_{s-1}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-1}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{0}{\bm{z}}_{s-1}\right)$ $\displaystyle G(II)$ $\displaystyle\doteq$
$\displaystyle\frac{1}{n(n-1)}\sum_{s\not=t}^{T}\left({\bm{z}}_{s}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-1}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{0}{\bm{z}}_{s-1}+{\bm{z}}_{s-1}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-1}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{0}{\bm{z}}_{s-1}\right.$
$\displaystyle+{\bm{z}}_{s-1}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-1}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{1}{\bm{z}}_{s-2}+{\bm{z}}_{s-1}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-2}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{0}{\bm{z}}_{s-1}$ $\displaystyle+{\bm{z}}_{s}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-2}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{0}{\bm{z}}_{s-1}+{\bm{z}}_{s}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-1}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{1}{\bm{z}}_{s-2}$ $\displaystyle\left.+{\bm{z}}_{s}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-2}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{1}{\bm{z}}_{s-2}+{\bm{z}}_{s-1}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-2}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{1}{\bm{z}}_{s-2}\right)$ $\displaystyle G(III)$
$\displaystyle\doteq$
$\displaystyle\frac{1}{n(n-1)}\sum_{s\not=t}^{T}\left({\bm{z}}_{s}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-1}^{\top}{\bf
A}_{0}^{\top}{\bf A}_{1}{\bm{z}}_{s-2}+{\bm{z}}_{s-1}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-2}^{\top}{\bf
A}_{1}^{\top}{\bf A}_{0}{\bm{z}}_{s-1}\right.$
$\displaystyle\left.+{\bm{z}}_{s}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{1}{\bm{z}}_{t-1}{\bm{z}}_{t-2}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{0}{\bm{z}}_{s-1}+{\bm{z}}_{s-1}^{\top}{\bf A}_{1}^{\top}{\bf
A}_{0}{\bm{z}}_{t}{\bm{z}}_{t-1}^{\top}{\bf A}_{0}^{\top}{\bf
A}_{1}{\bm{z}}_{s-2}\right).$

After some tedious algebra, we have

$\displaystyle\mathbb{E}\\{G(I)\\}=\mathrm{tr}(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1}),~{}\mathbb{E}\\{G(II)\\}=0,~{}\mathbb{E}\\{G(III)\\}=\frac{2}{T}\mathrm{tr}^{2}(\tilde{{\bm{\Sigma}}}_{01})$

and

$\displaystyle\mathrm{var}\\{G(I)\\}=$
$\displaystyle\frac{2}{T^{2}}\mathrm{tr}^{2}(\tilde{\bm{\Sigma}}_{0}^{2}+\tilde{\bm{\Sigma}}_{1}^{2})+\frac{6}{T^{2}}\mathrm{tr}^{2}(\tilde{\bm{\Sigma}}_{0}\tilde{\bm{\Sigma}}_{1})$
$\displaystyle+\frac{4}{T}\left[2\operatorname{tr}\left(\widetilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1}\right)^{2}+\left(\nu_{4}-3\right)\operatorname{tr}\left\\{D^{2}\left(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{1}\right)\right\\}\right]+r_{n},$
$\displaystyle\mathrm{var}\\{G(II)\\}=$
$\displaystyle\frac{8}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top})\mathrm{tr}(\tilde{\bm{\Sigma}}_{0}^{2}+\tilde{\bm{\Sigma}}_{1}^{2})+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{1})\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{0})$
$\displaystyle+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{0}+\tilde{{\bm{\Sigma}}}_{1})\left\\{\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}^{\top}\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{0})+\mathrm{tr}(\tilde{\bm{\Sigma}}_{01}\tilde{\bm{\Sigma}}_{01}^{\top}\tilde{\bm{\Sigma}}_{1})\right\\}$
$\displaystyle+\frac{16}{T^{2}}\mathrm{tr}(\tilde{\bm{\Sigma}}_{01})\left\\{\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{0}^{2}\tilde{{\bm{\Sigma}}}_{01}^{\top}\right)+\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{1}^{2}\tilde{{\bm{\Sigma}}}_{01}\right)+2\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0}\right)\right\\}$
$\displaystyle+\frac{4}{T}\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0}^{2}+\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{1}^{2}+2\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{0}\right)+r_{n},$
$\displaystyle\mathrm{var}\\{G(III)\\}=$
$\displaystyle\frac{4}{T}\operatorname{tr}\left(\widetilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}\right)+\frac{12}{T^{2}}\operatorname{tr}^{2}\left(\widetilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\right)$
$\displaystyle+\frac{16}{T^{2}}\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{01}\right)\operatorname{tr}\left(\tilde{{\bm{\Sigma}}}_{01}\tilde{{\bm{\Sigma}}}_{01}^{\top}\tilde{{\bm{\Sigma}}}_{01}^{\top}\right)+r_{n},$
$\displaystyle\mathrm{cov}\\{G(I),G(III)\\}=$
$\displaystyle\frac{4}{T^{2}}\operatorname{tr}^{2}\left(\tilde{{\bm{\Sigma}}}_{0}\tilde{{\bm{\Sigma}}}_{01}\right)+\frac{4}{T^{2}}\operatorname{tr}^{2}\left(\tilde{{\bm{\Sigma}}}_{1}\tilde{{\bm{\Sigma}}}_{01}\right)+r_{n},$
$\displaystyle\mathrm{cov}\\{G(I),G(II)\\}=$ $\displaystyle
r_{n},~{}~{}\mathrm{cov}\\{G(II),G(III)\\}=r_{n}.$

Similar to the proof of (24), we have

$\displaystyle\frac{T_{\operatorname{SUM}}-\mu_{1}}{\sigma_{S1}}\mathop{\rightarrow}\limits^{d}\mathcal{N}(0,1).$

$\square$

###  6.7 Proof of Theorem 6

First, we present some technical results for the proof of Theorem 6.

The following is a well-known formula for conditional distributions of
multivariate normal distributions; see, for example, p.12 from Muirhead
(1982).

######  Lemma 6.

Let $\mathbb{X}\sim\mathcal{N}(\bm{\mu},\mathbb{\Sigma})$ with
$\mathbb{\Sigma}$ being invertible. Partition $\mathbb{X},\bm{\mu}$ and
$\mathbb{\Sigma}$ as

$\displaystyle\mathbb{X}=\begin{pmatrix}\mathbb{X}_{1}\\\
\mathbb{X}_{2}\end{pmatrix},\ \ \ \
\mathbb{{\bm{\mu}}}=\begin{pmatrix}\mathbb{{\bm{\mu}}}_{1}\\\
\mathbb{{\bm{\mu}}}_{2}\end{pmatrix},\ \ \ \
\mathbb{\Sigma}=\begin{pmatrix}\mathbb{\Sigma}_{11}&\mathbb{\Sigma}_{12}\\\
\mathbb{\Sigma}_{21}&\mathbb{\Sigma}_{22}\end{pmatrix},$

where
$\mathbb{X}_{2}\sim\mathcal{N}(\mathbb{{\bm{\mu}}}_{2},\mathbb{\Sigma}_{22})$.
Set $\mathbb{\Sigma}_{22\cdot
1}=\mathbb{\Sigma}_{22}-\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{\Sigma}_{12}$.
Then
$\mathbb{X}_{2}-\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{X}_{1}\sim\mathcal{N}_{k}(\mathbb{{\bm{\mu}}}_{2}-\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{{\bm{\mu}}}_{1},\mathbb{\Sigma}_{22\cdot
1})$ and is independent of $\mathbb{X}_{1}$.

######  Lemma 7.

For any $x\in\mathbb{R}$ and $y\in\mathbb{R}$, define
$A_{p}=\\{\frac{T_{\operatorname{SUM}}}{\sigma_{S}}\leq x\\}$ and
$l_{p}=(2\log N-\log\log N+y)^{1/2}$ and $B_{i}=\\{|Z_{i}|>l_{p}\\}.$ Then,
for each $1\leq d\leq p$,

$\displaystyle\sum_{1\leq i_{1}<\cdots<i_{d}\leq
p}\big{|}\textrm{P}(A_{p}B_{i_{1}}\cdots
B_{i_{d}})-\textrm{P}(A_{p})\cdot\textrm{P}(B_{i_{1}}\cdots
B_{i_{d}})\big{|}\to 0$

as $p\to\infty$.

Proof. The argument is divided into two steps.

Step 1: appealing independence from normal distributions.
Note that
$(\varepsilon_{t1},\cdots,\varepsilon_{tp})^{\top}\sim\mathcal{N}(\mathbb{0},\mathbb{\Sigma})$.
Take $\mathbb{X}_{t1}=(\varepsilon_{t1},\cdots,\varepsilon_{td})^{\top}$ and
$\mathbb{X}_{t2}=(\varepsilon_{t,d+1},\cdots,\varepsilon_{tp})^{\top}$. Recall
the notation in Lemma 6. Write
$\mathbb{X}_{t2}=\mathbb{U}_{t}+\mathbb{V}_{t}$, where
$\mathbb{U}_{t}=\mathbb{X}_{t2}-\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{X}_{t1}\sim\mathcal{N}(\mathbb{0},\mathbb{\Sigma}_{22\cdot
1})$ and
$\mathbb{V}_{t}=\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{X}_{t1}\sim\mathcal{N}(\mathbb{0},\mathbb{\Sigma}_{21}\mathbb{\Sigma}_{11}^{-1}\mathbb{\Sigma}_{12})$.
By Lemma 6,

$\mathbb{U}_{t}\ \mbox{and}\ \\{\varepsilon_{t1},\cdots,\varepsilon_{td}\\}\
\mbox{are independent}.$ (29)

Write

$\displaystyle T_{\operatorname{SUM}}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}\bm{\varepsilon}_{t}^{\top}\bm{\varepsilon}_{s}\bm{\varepsilon}_{t+l}^{\top}\bm{\varepsilon}_{s+l}$
$\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}({\bm{X}}_{t1}^{\top}{\bm{X}}_{s1}+{\bm{X}}_{t2}^{\top}{\bm{X}}_{s2})({\bm{X}}_{t+l,1}^{\top}{\bm{X}}_{s+l,1}+{\bm{X}}_{t+l,2}^{\top}{\bm{X}}_{s+l,2})$
$\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}({\bf
U}_{t}^{\top}{\bf U}_{s}+{\bf U}_{t}^{\top}{\bm{V}}_{s}+{\bf
U}_{s}{\bm{V}}_{t}+{\bm{V}}_{t}{\bm{V}}_{s}+{\bm{X}}_{t1}^{\top}{\bm{X}}_{s1})$
$\displaystyle\times({\bf U}_{t+l}^{\top}{\bf U}_{s+l}+{\bf
U}_{t+l}^{\top}{\bm{V}}_{s+l}+{\bf
U}_{s+l}{\bm{V}}_{t+l}+{\bm{V}}_{t+l}{\bm{V}}_{s+l}+{\bm{X}}_{t+l,1}^{\top}{\bm{X}}_{s+l,1})$
$\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf U}_{t+l}^{\top}{\bf
U}_{s+l}+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf U}_{t+l}^{\top}{\bm{V}}_{s+l}$
$\displaystyle+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf
U}_{s}{\bm{V}}_{t+l}^{\top}{\bm{V}}_{s+l}+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bm{X}}_{t+l,1}^{\top}{\bm{X}}_{s+l,1}$
$\displaystyle+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bm{V}}_{s}{\bf U}_{t+l}^{\top}{\bf
U}_{s+l}+\frac{4}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bm{V}}_{s}{\bf U}_{t+l}^{\top}{\bm{V}}_{s+l}$
$\displaystyle+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bm{V}}_{s}{\bm{V}}_{t+l}^{\top}{\bm{V}}_{s+l}+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bm{V}}_{s}{\bm{X}}_{t+l,1}^{\top}{\bm{X}}_{s+l,1}$
$\displaystyle+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bm{X}}_{t1}^{\top}{\bm{X}}_{s1}{\bf
U}_{t+l}^{\top}{\bf
U}_{s+l}+\frac{4}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bm{X}}_{t1}^{\top}{\bm{X}}_{s1}{\bf
U}_{t+l}^{\top}{\bm{V}}_{s+l}$
$\displaystyle+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bm{X}}_{t1}^{\top}{\bm{X}}_{s1}{\bm{V}}_{t+l}^{\top}{\bm{V}}_{s+l}+\frac{2}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bm{X}}_{t1}^{\top}{\bm{X}}_{s1}{\bm{X}}_{t+l,1}^{\top}{\bm{X}}_{s+l,1}$
$\displaystyle\doteq$
$\displaystyle\frac{1}{n(n-1)}\sum_{l=1}^{K}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf U}_{t+l}^{\top}{\bf
U}_{s+l}+\sum_{q=1}^{11}\Theta_{q}$ $\displaystyle\doteq$ $\displaystyle
S_{p}+R_{p}.$

Next, we will show that, for any $d\geq 1$ and $\iota>0$, there exists
$t=t_{p}>0$ with $\lim_{N\to\infty}t_{p}=\infty$ and integer $p_{0}\geq 1$
such that

$\displaystyle\textrm{P}(|\Theta_{q}|\geq\iota\sigma_{S})\leq\frac{1}{p^{t}}$
(30)

as $p\geq p_{0}$. Here we only consider $\Theta_{1}$. The proof of the other
parts are similar to $\Theta_{1}$.

By the decomposition, we know that $\\{{\bm{V}}_{t}\\}_{t=1}^{n}$ is
independent of $\\{{\bf U}_{t}\\}_{t=1}^{n}$. Thus, conditional on $\\{{\bf
U}_{t}\\}_{t=1}^{n}$, $\Theta_{1}$ has the normal distribution. Hence, we have

$\displaystyle\operatorname{P}(|\Theta_{1}|\geq\iota\sigma_{S})\leq\sum_{l=1}^{K}\operatorname{P}\left(\left|\frac{2}{n(n-1)}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf
U}_{t+l}^{\top}{\bm{V}}_{s+l}\right|\geq\iota\sigma_{S}/K\right)$
$\displaystyle=\sum_{l=1}^{K}\mathbb{E}\left(\mathbb{E}\left[\mathbb{I}\left\\{\left|\frac{2}{n(n-1)}\underset{t\not=s}{\sum\sum}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf
U}_{t+l}^{\top}{\bm{V}}_{s+l}\right|\geq\iota\sigma_{S}/K\right\\}|\\{{\bf
U}_{t}\\}_{t=1}^{n}\right]\right)$
$\displaystyle=2\sum_{l=1}^{K}\mathbb{E}\left\\{1-\Phi\left(\frac{\iota\sigma_{S}}{K\tilde{\sigma}_{l}}\right)\right\\}\simeq\sum_{l=1}^{K}\mathbb{E}\left\\{\frac{2}{\sqrt{2\pi}K^{-1}\iota\tilde{\sigma}_{l}^{-1}\sigma_{S}}e^{-(K^{-1}\iota\tilde{\sigma}_{l}^{-1}\sigma_{S})^{2}/2}\right\\},$

where

$\displaystyle\tilde{\sigma}^{2}_{l}=\frac{4}{n^{2}(n-1)^{2}}\sum_{t,m\not=s}{\bf
U}_{t}^{\top}{\bf U}_{s}{\bf
U}_{t+l}^{\top}{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12}{\bf
U}_{m+l}{\bf U}_{m}^{\top}{\bf U}_{s}.$

Here, $a_{n}\simeq b_{n}$ denotes that $a_{n}/b_{n}\to 1$. Similar to the
proof of Proposition 1, we have

$\displaystyle\frac{\tilde{\sigma}^{2}_{l}}{\frac{4}{n(n-1)}\mathrm{tr}({\bm{\Sigma}}_{22\cdot
1}^{2})\mathrm{tr}({\bm{\Sigma}}_{22\cdot
1}\cdot{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})}\mathop{\rightarrow}\limits^{p}1.$

Thus,

$\displaystyle
K^{-1}\iota\tilde{\sigma}_{l}^{-1}\sigma_{S}\mathop{\rightarrow}\limits^{p}\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}^{2})\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}\cdot{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})}$

and

$\displaystyle\operatorname{tr}\left\\{\bm{\Sigma}_{22\cdot
1}\cdot\left(\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}\right)\right\\}\leq\lambda_{\max}\left(\bm{\Sigma}_{22\cdot
1}\right)\cdot\operatorname{tr}\left(\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}\right)$
$\displaystyle\leq$
$\displaystyle\lambda_{\max}(\bm{\Sigma})\cdot\operatorname{tr}\left(\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}\right)\leq\lambda_{\max}(\bm{\Sigma})\lambda_{\max}\left({\bm{\Sigma}}_{11}^{-1}\right)\cdot\operatorname{tr}\left({\bm{\Sigma}}_{12}{\bm{\Sigma}}_{21}\right)$
$\displaystyle=$
$\displaystyle\lambda_{\max}(\bm{\Sigma})\frac{1}{\lambda_{\min}\left({\bm{\Sigma}}_{11}\right)}\cdot\operatorname{tr}\left({\bm{\Sigma}}_{12}{\bm{\Sigma}}_{21}\right)\leq{dM_{p}\lambda_{\max}(\bm{\Sigma})}.$

In fact, in the above, we use the assertion
$\lambda_{\min}\left({\bm{\Sigma}}_{11}\right)$ is bounded by Condition (C2)
and the fact that

$\operatorname{tr}\left(\bm{\Sigma}_{12}\bm{\Sigma}_{21}\right)=\sum_{i=1}^{d}\sum_{j=d+1}^{p}\sigma_{ij}^{2}\leq
dM_{p}.$

Additionally, $\bm{\Sigma}_{22}=\bm{\Sigma}_{22\cdot
1}+\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}$ and all three
matrices are non-negative definite. Hence, we have
$\mathrm{tr}({\bm{\Sigma}}^{2})\geq\mathrm{tr}({\bm{\Sigma}}^{2}_{22\cdot
1})$. Thus,

$\displaystyle\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}^{2})\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}\cdot{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})}\geq\frac{\iota}{K}\frac{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2})}{\sqrt{dM_{p}\lambda_{\max}({\bm{\Sigma}})}}.$

Then,

$\displaystyle\sum_{l=1}^{K}\mathbb{E}\left\\{\frac{2}{\sqrt{2\pi}K^{-1}\iota\tilde{\sigma}_{l}^{-1}\sigma_{S}}e^{-(K^{-1}\iota\tilde{\sigma}_{l}^{-1}\sigma_{S})^{2}/2}\right\\}$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2})}{\sqrt{dM_{p}\lambda_{\max}({\bm{\Sigma}})}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{dM_{p}\lambda_{\max}({\bm{\Sigma}})}\right\\}\leq
p^{-t_{p}}$

by Condition (C6) with $t_{p}=\frac{\iota^{2}}{2K^{2}}(\log p)^{\gamma-1}$.
Note that $\mathrm{tr}({\bm{\Sigma}}^{2}_{11})\leq C_{d}$ is bounded.

Similarly, it can be proved that for the other parts
$\Theta_{2},\cdots,\Theta_{11}$,

$\displaystyle\mathrm{P}(|\Theta_{2}|\geq\iota\sigma_{S})$ $\displaystyle\leq$
$\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2}_{22\cdot
1})\mathrm{tr}^{1/2}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}}}\exp\left[-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}({\bm{\Sigma}}^{2}_{22\cdot
1})\mathrm{tr}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2})}{dM_{p}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{d^{2}M^{2}_{p}}\right\\}\leq
p^{-t_{p}},$ $\displaystyle\mathrm{P}(|\Theta_{3}|\geq\iota\sigma_{S})$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2}_{22\cdot
1})\mathrm{tr}^{1/2}({\bm{\Sigma}}_{11}^{2})}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}({\bm{\Sigma}}^{2}_{22\cdot
1})\mathrm{tr}({\bm{\Sigma}}_{11}^{2})}\right\\}$ $\displaystyle\leq$
$\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}^{1/2}({\bm{\Sigma}}^{2})}{C_{d}^{1/2}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{C_{d}}\right\\}\leq
p^{-t_{p}},$ $\displaystyle\mathrm{P}(|\Theta_{4}|\geq\iota\sigma_{S})$
$\displaystyle\simeq\mathrm{P}(|\Theta_{1}|\geq\iota\sigma_{S}),~{}~{}\mathrm{P}(|\Theta_{5}|\geq\iota\sigma_{S})\simeq\mathrm{P}(|\Theta_{2}|\geq\iota\sigma_{S}),$
$\displaystyle\mathrm{P}(|\Theta_{6}|\geq\iota\sigma_{S})$ $\displaystyle\leq$
$\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})\mathrm{tr}^{1/2}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}}}$
$\displaystyle\times\exp\left[-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}({\bm{\Sigma}}_{22\cdot
1}{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})\mathrm{tr}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\left\\{d^{3/2}K^{3/2}_{p}\lambda_{\max}({\bm{\Sigma}})\right\\}^{1/2}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{d^{3}M^{3}_{p}\lambda_{\max}({\bm{\Sigma}})}\right\\}\leq
p^{-t_{p}},$ $\displaystyle\mathrm{P}(|\Theta_{7}|\geq\iota\sigma_{S})$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}({\bm{\Sigma}}_{22\cdot
1}{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})\mathrm{tr}^{1/2}({\bm{\Sigma}}_{11}^{2})}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}({\bm{\Sigma}}_{22\cdot
1}{\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})\mathrm{tr}({\bm{\Sigma}}_{11}^{2})}\right\\}$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\left\\{C_{d}dM_{p}\lambda_{\max}({\bm{\Sigma}})\right\\}^{1/2}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{C_{d}dM_{p}\lambda_{\max}({\bm{\Sigma}})}\right\\}\leq
p^{-t_{p}},$ $\displaystyle\mathrm{P}(|\Theta_{8}|\geq\iota\sigma_{S})$
$\displaystyle\simeq\mathrm{P}(|\Theta_{3}|\geq\iota\sigma_{S}),~{}~{}\mathrm{P}(|\Theta_{9}|\geq\iota\sigma_{S})\simeq\mathrm{P}(|\Theta_{7}|\geq\iota\sigma_{S}),$
$\displaystyle\mathrm{P}(|\Theta_{10}|\geq\iota\sigma_{S})$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{1/2}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}\mathrm{tr}^{1/2}({\bm{\Sigma}}_{11}^{2})}}\exp\left[-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}\left\\{({\bm{\Sigma}}_{21}{\bm{\Sigma}}_{11}^{-1}{\bm{\Sigma}}_{12})^{2}\right\\}\mathrm{tr}({\bm{\Sigma}}_{11}^{2})}\right]$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{C_{d}^{1/2}dM_{p}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{C_{d}d^{2}M^{2}_{p}}\right\\}\leq
p^{-t_{p}},$ $\displaystyle\mathrm{P}(|\Theta_{11}|\geq\iota\sigma_{S})\leq$
$\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{\mathrm{tr}({\bm{\Sigma}}_{11}^{2})}}\exp\left\\{-\frac{\iota}{K}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{\mathrm{tr}^{2}({\bm{\Sigma}}_{11}^{2})}\right\\}$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{2\pi}}\frac{2}{\frac{\iota}{K}\frac{\mathrm{tr}({\bm{\Sigma}}^{2})}{C_{d}^{1/2}}}\exp\left\\{-\frac{\iota^{2}}{K^{2}}\frac{\mathrm{tr}^{2}({\bm{\Sigma}}^{2})}{C_{d}}\right\\}\leq
p^{-t_{p}}.$

Thus, we have

$\displaystyle\textrm{P}(|R_{p}|\geq\iota\sigma_{S})\leq\frac{1}{p^{t_{p}}}.$
(31)

Now, for clarity, we will revise the definition of $A_{p}$ as follows

$\displaystyle A_{p}(x)=\Big{\\{}\frac{T_{\operatorname{SUM}}}{\sigma_{S}}\leq
x\Big{\\}},\ \ x\in\mathbb{R},$

for $p\geq 1$. Due to the fact that $T_{\operatorname{SUM}}=S_{p}+R_{p}$, we
see that

$\displaystyle\operatorname{P}\\{A_{p}(x)B_{1}\cdots B_{d}\\}$
$\displaystyle\leq$ $\displaystyle\mathrm{P}\Big{\\{}A_{p}(x)B_{1}\cdots
B_{d},\ \frac{|R_{p}|}{\sigma_{S}}<\iota\Big{\\}}+\frac{1}{p^{t}}$
$\displaystyle\leq$
$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq x+\iota,\
B_{1}\cdots B_{d}\Big{)}+\frac{1}{p^{t}}$ $\displaystyle=$
$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq
x+\iota\Big{)}\cdot P\big{(}B_{1}\cdots B_{d}\big{)}+\frac{1}{p^{t}}$

by the independence appeared in (29). Hence,

$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq x+\iota\Big{)}$
$\displaystyle\leq$
$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq x+\iota,\
\frac{|R_{p}|}{\sigma_{S}}<\iota\Big{)}+\frac{1}{p^{t}}$ $\displaystyle\leq$
$\displaystyle\mathrm{P}\Big{\\{}\frac{1}{\sigma_{S}}(S_{p}+R_{p})\leq
x+2\iota\Big{\\}}+\frac{1}{p^{t}}\leq\textrm{P}\big{\\{}A_{p}(x+2\iota)\big{\\}}+\frac{1}{p^{t}}.$

Combine the two inequalities to get

$\displaystyle\operatorname{P}\\{A_{p}(x)B_{1}\cdots
B_{d}\\}\leq\operatorname{P}\big{\\{}A_{p}(x+2\iota)\big{\\}}\cdot\operatorname{P}\big{(}B_{1}\cdots
B_{d}\big{)}+\frac{2}{p^{t}}.$ (32)

Similarly,

$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq x-\iota,\
B_{1}\cdots B_{d}\Big{)}$ $\displaystyle\leq$
$\displaystyle\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq
x-\iota,B_{1}\cdots
B_{d},\frac{|R_{p}|}{\sigma_{S}}<\iota\Big{)}+\frac{1}{p^{t}}\leq\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq
x,\ B_{1}\cdots B_{d}\Big{)}+\frac{1}{p^{t}}.$

By the independence from (29),

$\displaystyle\mathrm{P}\\{A_{p}(x)B_{1}\cdots
B_{d}\\}\geq\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq
x-\iota\Big{)}\cdot\mathrm{P}(B_{1}\cdots B_{d})-\frac{1}{p^{t}}.$

Furthermore,

$\mathrm{P}\Big{(}\frac{T_{\operatorname{SUM}}}{\sigma_{S}}\leq
x-2\iota\Big{)}\leq\mathrm{P}\Big{(}\frac{T_{\operatorname{SUM}}}{\sigma_{S}}\leq
x-2\iota,\
\frac{|R_{p}|}{\sigma_{S}}<\iota\Big{)}+\frac{1}{p^{t}}\leq\mathrm{P}\Big{(}\frac{S_{p}}{\sigma_{S}}\leq
x-\iota\Big{)}+\frac{1}{p^{t}},$

where the fact $T_{\operatorname{SUM}}=S_{p}+R_{p}$ is used again. Combining
the above two inequalities, we get

$\displaystyle\mathrm{P}\\{A_{p}(x)B_{1}\cdots
B_{d}\\}\geq\mathrm{P}\\{A_{p}(x-2\iota)\\}\cdot\mathrm{P}(B_{1}\cdots
B_{d})-\frac{2}{p^{t}}.$

This together with (32) concludes

$\displaystyle\left|\mathrm{P}\\{A_{p}(x)B_{1}\cdots
B_{d}\\}-\mathrm{P}\\{A_{p}(x)\\}\cdot\mathrm{P}(B_{1}\cdots
B_{d})\right|\leq\Delta_{p,\iota}\cdot\mathrm{P}(B_{1}\cdots
B_{d})+\frac{2}{p^{t}}$ (33)

as $p\geq p_{0}$, where

$\displaystyle\Delta_{p,\iota}$ $\displaystyle\doteq$
$\displaystyle|\mathrm{P}\\{A_{p}(x)\\}-\mathrm{P}\\{A_{p}(x+2\iota)\\}|+|\mathrm{P}\\{A_{p}(x)\\}-\mathrm{P}\\{A_{p}(x-2\iota)\\}|$
$\displaystyle=$
$\displaystyle\mathrm{P}\\{A_{p}(x+2\iota)\\}-\mathrm{P}\\{A_{p}(x-2\iota)\\}$

since $\mathrm{P}\\{A_{p}(x)\\}$ is increasing in $x\in\mathbb{R}.$ An
important observation is that the derivation of (30) is based on three key
facts: inequality (30), the identity $T_{\operatorname{SUM}}=S_{p}+R_{p}$ and
the fact $\mathbb{U}_{t}$ and $\\{\varepsilon_{t1},\cdots,\varepsilon_{td}\\}$
are independent from (29).

Thus, the three corresponding key facts aforementioned also hold for the
quantities related to $\Lambda=\\{i_{1},\cdots,i_{d}\\}$. Therefore, similar
to the derivation of (33), we have

$\displaystyle\big{|}\mathrm{P}\\{A_{p}(x)B_{i_{1}}\cdots
B_{i_{d}}\\}-\mathrm{P}\\{A_{p}(x)\\}\cdot\mathrm{P}(B_{i_{1}}\cdots
B_{i_{d}})\big{|}$ $\displaystyle\leq$
$\displaystyle\Delta_{p,\iota}\cdot\mathrm{P}(B_{i_{1}}\cdots
B_{i_{d}})+\frac{2}{p^{t}}$

as $p\geq p_{0}$. As a result,

$\displaystyle\zeta(p,d)\doteq$ $\displaystyle\sum_{1\leq
i_{1}<\cdots<i_{d}\leq p}\big{|}\mathrm{P}\\{A_{p}(x)B_{i_{1}}\cdots
B_{i_{d}}\\}-\mathrm{P}\\{A_{p}(x)\\}\cdot\mathrm{P}(B_{i_{1}}\cdots
B_{i_{d}})\big{|}$ $\displaystyle\leq$ $\displaystyle\sum_{1\leq
i_{1}<\cdots<i_{d}\leq
N}\Big{\\{}\Delta_{p,\iota}\cdot\mathrm{P}(B_{i_{1}}\cdots
B_{i_{d}})+\frac{2}{p^{t}}\Big{\\}}$ $\displaystyle\leq$
$\displaystyle\Delta_{p,\iota}\cdot H(d,p)+\binom{p}{d}\cdot\frac{2}{p^{t}},$
(34)

where

$\displaystyle H(d,N)\doteq\sum_{1\leq i_{1}<\cdots<i_{d}\leq