Rank Based Tests for High Dimensional White Noise

Dachuan Chen, Fengyi Song and Long Feng

Nankai University

> Abstract: The development of high-dimensional white noise test is important
> in both statistical theories and applications, where the dimension of the
> time series can be comparable to or exceed the length of the time series.
> This paper proposes several distribution-free tests using the rank based
> statistics for testing the high-dimensional white noise, which are robust to
> the heavy tails and do not quire the finite-order moment assumptions for the
> sample distributions. Three families of rank based tests are analyzed in
> this paper, including the simple linear rank statistics, non-degenerate
> U-statistics and degenerate U-statistics. The asymptotic null distributions
> and rate optimality are established for each family of these tests. Among
> these tests, the test based on degenerate U-statistics can also detect the
> non-linear and non-monotone relationships in the autocorrelations. Moreover,
> this is the first result on the asymptotic distributions of rank correlation
> statistics which allowing for the cross-sectional dependence in high
> dimensional data.
>
> Key words and phrases: Key Words: White noise test, Serial correlation, High
> dimensionality, Simple linear rank statistics, Non-degenerate U-statistics,
> Degenerate U-statistics.

##  1 Introduction

The hypothesis test for white noise is a critical methodology in statistical
inference and modeling. It is necessary in diagnosis checking for the linear
regression and time series modeling. There has been a vast increase in the
amount of high-dimensional data available in recent years, which has received
increasing attention from statisticians. The existence of such high-
dimensional data is widespread, including the areas of genomics, neuroscience,
finance, economics and so on. This brings additional difficulties for the
problem of diagnosis checking, which means that in the theoretical development
of the test for high-dimensional white noise, the dimension of the time series
can be comparable to or exceed the length of the time series.

For the white noise tests designed for univariate time series, many commonly
used methodologies are well documented in Li (2004). The alternative
hypothesis of these tests can be grouped into two different classes: (i)
specified alternative in form of some explicit parametric model; (ii)
completely unspecified alternative, which means that the departure from white
noise can be arbitrary. It is well known that likelihood based tests are more
powerful than the omnibus tests under the first class of the alternatives,
see, e.g. Chang et al. (2017). Under the second class of alternatives, the
Box-Pierce portmanteau test and its variations are most popular because of its
ease of use in practice and motivate the white noise tests for multivariate
time series such as Hosking (1980) and Li and McLeod (1981). Specially, the
Ljung–Box test is a type of statistical test of whether any of a group of
autocorrelations of a time series are different from zero. Instead of testing
randomness at each distinct lag, it tests the “overall” randomness based on a
number of lags. These tests enjoy the theoretical benefits of asymptotically
distribution-free and $\chi^{2}$-distributed properties under null hypothesis,
see, e.g. Li (2004) and Lütkepohl (2005).

There are also some white noise tests constructed for the multivariate time
series which assuming that the dimension of the times series is smaller than
the length of the time series in asymptotics, see, e.g., Hosking (1980) and Li
and McLeod (1981). However, the existing literature suggests that these tests
suffer from the slow convergence to their asymptotic null distributions, see
Li et al. (2019). This fact calls for the more efficient testing methodologies
for multivariate time series, or even high-dimensional time series.

Several omnibus tests for high-dimensional white noise have been developed in
recent years, see, e.g., Chang et al. (2017), Li et al. (2019), Tsay (2020)
and Feng et al. (2022b). Among these existing theories, the tests proposed in
Chang et al. (2017), Li et al. (2019) and Feng et al. (2022b) are
distribution-dependent, while the test in Tsay (2020) is distribution-free.
Chang et al. (2017) developed a max-type test for this purpose based on the
maximum absolute auto-correlations and cross-correlations of the component
series. Li et al. (2019) proposed a sum-type test for high dimensional white
noise by summing up the squared singular values of the first several lagged
sample auto-covariance matrices. In general, the max-type test can only work
well under the sparse alternatives where only a few elements in the auto-
correlations are nonzero. In contrast, the sum-type test can only work well
under the dense alternatives. To test the high dimensional white noise, Feng
et al. (2022b) show the asymptotic independence between the max-type test
statistic and a new sum-type test statistic. Based on this theoretical result,
this paper constructed the Fisher’s combination test which is robust to both
sparse and dense alternatives. As a distribution-free approach, Tsay (2020)
developed the high-dimensional white noise test based on the Spearman’s rank
correlation and the theory of extreme values.

More accurately, in this paper we consider the following hypothesis testing
problem. Let $\boldsymbol{\varepsilon}_{t}$ be a $p$-dimensional weakly
stationary time series with mean zero. We want to test the following
hypothesis:

$\displaystyle
H_{0}:\\{\boldsymbol{\varepsilon}_{t}\\}~{}{\text{i}s~{}white~{}noise~{}versus}~{}H_{1}:\\{\boldsymbol{\varepsilon}_{t}\\}~{}is~{}not~{}white~{}noise$
(1.1)

In this paper, we said a time series $x_{1},\cdots,x_{T}$ are white noise if
they are all independent and identically distributed. So, under the null
hypothesis, $\boldsymbol{\varepsilon}_{t+k}$ is independent of
$\boldsymbol{\varepsilon}_{t}$ for all $k>0$. Here the dimension of the time
series $p$ is comparable to or even larger than the sample size $n$.

In this paper, we develop the rank based tests for testing the high-
dimensional white noise, which are distribution-free. The proposed tests are
robust to the heavy tails and do not require the finite-order moment
assumptions or any tail assumptions for the sample distribution. There are
three families of rank based tests investigated in this paper, including the
simple linear rank statistics, non-degenerate U-statistics and degenerate
U-statistics, with the examples of Spearman’s rho, Kendall’s tau, Hoeffding’s
$D$, Blum-Kiefer-Rosenblatt’s $R$ and Bergsma-Dassios-Yanagimoto’s $\tau^{*}$.
Among these tests, simple linear rank statistics and non-degenerate
U-statistics can only work well with the linear or monotone relationships in
autocorrelations. In contrast, the degenerate U-statistics can also work well
with the non-linear and non-monotone relationships in autocorrelations. As the
theoretical results of this paper, we have established the asymptotic null
distribution, the power analysis and the rate optimality in terms of power for
each family of the rank based test statistics.

Because this paper shows one possible application of the rank correlation
statistics in high-dimensional data analysis, we here provide a brief
literature review for the rank correlation statistics and point out the
theoretical contribution of this paper. Han et al. (2017) proposed the rank
based tests based on the simple linear rank statistics and non-degenerate
U-statistics for testing the mutual independence among all elements in the
high-dimensional random vectors. Drton et al. (2020) proposed the hypothesis
test based on the degenerate U-statistics with the same purpose as Han et al.
(2017). As mentioned earlier, Tsay (2020) applied the Spearman’s rank
correlation to the test of high-dimensional white noise. However, the
asymptotic distributions of the rank correlation statistics in these three
existing literature are all derived based on the assumption of cross-sectional
independence in high-dimensional data. Therefore, as the theoretical
contribution of our result, this is the first paper in existing literature
which established the asymptotic distribution of the rank correlation
statistics without assuming the cross-sectional independence.

The main contributions of this paper are summarized as follows.

  1. 1.

We develop the rank based tests for testing the high dimensional white noise,
which are distribution free. Our test are robust to the heavy tails and do not
require the finite-order moment assumptions.

  2. 2.

Besides the simple linear rank statistics and non-degenerate U-statistics, we
also develop the tests for the degenerate U-statistics, which are very useful
to detect the non-linear and non-monotone relationships in autocorrelations.
Limiting null distributions and the rate optimality in terms of power of these
three families of tests are established in this paper.

  3. 3.

In the existing literature concerning the asymptotic distribution of rank
correlation statistics, this paper is the first one on this topic which
allowing for the cross-sectional dependence in the high-dimensional data. In
contrast, the other existing results are all based on the assumption of cross-
sectional independence of the data, see, e.g., Han et al. (2017), Drton et al.
(2020) and Tsay (2020).

This paper is organized as follows. Section 2 proposes the theoretical results
about three families of distribution-free test statistics, including the
simple linear rank statistics, non-degenerate U-statistics and degenerate
U-statistics. The limiting null distributions of these tests are derived and
their rate-optimality in terms of power is also analyzed. Section 3 shows the
empirical sizes and the power comparison of the proposed test statistics based
on Monte Carlo simulation. Section 4 concludes this paper and discusses
several possible directions for the research in the future. All mathematical
proofs of the theoretical results in this paper are collected in supplementary
material.  In the supplementary material, we also consider high dimensional
white noise test based on Chatterjee’s rank Correlation (Chatterjee, 2021) and
$L$-statistics with the above three-type rank based correlations (Chang et
al., 2023).

##  2 Rank based tests

In this section, we state the theoretical results for three families of rank
based methodologies for testing the high-dimensional white noise, including
simple linear rank statistics, non-degenerate U-statistics and degenerate
U-statistics.

###  2.1 Simple linear rank statistics

First, we restate the definition of relative ranks in Han et al. (2017).
Consider the dependence between
$\\{(\varepsilon_{1,i},\varepsilon_{k+1,j}),\cdots,(\varepsilon_{n-k,i},\varepsilon_{n,j})\\}$
for any two entries $i,j\in\\{1,\ldots,p\\}$. Let $Q_{n-k,t}^{i}(k)$ be the
rank of $\varepsilon_{t,i}$ in
$\left\\{\varepsilon_{1,i},\ldots,\varepsilon_{n-k,i}\right\\}$ and let
$\tilde{Q}_{n-k,t+k}^{j}(k)$ be the rank of $\varepsilon_{t+k,j}$ in
$\\{\varepsilon_{k+1,j},\cdots,\varepsilon_{n,j}\\}$. Let
$R_{n-k,t+k}^{ij}(k)$ be the relative rank of $\varepsilon_{t+k,j}$ compared
to $\varepsilon_{t,i}$; that is,
$R_{n-k,t+k}^{ij}(k)\equiv\tilde{Q}_{n-k,t^{\prime}+k}^{j}(k)$ subject to the
constraint that $Q_{n-k,t^{\prime}}^{i}(k)=t$ for $t=1,\cdots,n-k$.

The first family includes tests based on simple linear rank statistics of the
form

$V_{ij}(k)\equiv(n-k)^{1/2}\sum_{t=1}^{n-k}c_{n-k,t}g\left\\{R_{n-k,t+k}^{ij}(k)/(n-k+1)\right\\}\quad(i,j\in\\{1,\ldots,p\\})$

where $\left\\{c_{n-k,t},t=1,\ldots,n-k\right\\}$ form an array of constants
called the regression constants and $g(\cdot)$ is a Lipschitz function called
the score function. We assume $\sum_{t=1}^{n-k}c_{n-k,t}^{2}>0$ to avoid
triviality. It is immediately clear that Spearman’s rho belongs to the family
of simple linear rank statistics. To accommodate tests of high-dimensional
white noise, we further pose the alignment assumption

$c_{n-k,t}=(n-k)^{-1}f\\{t/(n-k+1)\\}$

where $f(\cdot)$ is a Lipschitz function. Under this assumption, the simple
linear rank statistic is a general measure of the agreement between the ranks
of two sequences. The Spearman’s rho belongs to the family of simple linear
rank statistics with $g(x)=f(x)=x-\frac{1}{2}$.

Under $H_{0},$ the distribution of $V_{ij}(k)$ is irrelevant to the specific
distribution of $\varepsilon_{t}$ for all $i,j\in\\{1,\ldots,p\\}.$
Accordingly, the mean and variance of $V_{ij}(k)$ are calculable without
knowing the true distribution. Let $E_{H_{0}}(\cdot)$ and
$\operatorname{var}_{H_{0}}(\cdot)$ be the expectation and variance of a
certain statistic under $H_{0}.$ We have

$\displaystyle
E_{H_{0}}\left(V_{ij}(k)\right)=(n-k)^{1/2}\bar{g}_{n-k}\sum_{t=1}^{n-k}c_{n-k,t},$
(2.2)
$\displaystyle\sigma_{V}^{2}=\operatorname{var}_{H_{0}}\left(V_{ij}(k)\right)=\frac{n-k}{n-k-1}\sum_{t=1}^{n-k}\left[g\\{i/(n-k+1)\\}-\bar{g}_{n-k}\right]^{2}\sum_{t=1}^{n-k}\left(c_{n-k,t}-\bar{c}_{n-k}\right)^{2}$
(2.3)

where $\bar{g}_{n-k}\equiv(n-k)^{-1}\sum_{t=1}^{n-k}g\\{t/(n-k+1)\\}$ is the
sample mean of $g\left\\{R_{n-k,t}^{ij}/(n-k+1)\right\\}(t=1,\ldots,n-k)$ and
$\bar{c}_{n-k}=(n-k)^{-1}\sum_{t=1}^{n-k}c_{n-k,t}$. Based on
$\left\\{V_{ij}(k),1\leqslant i,j\leqslant p,1\leq k\leq K\right\\},$ we
propose the following statistic for testing $H_{0}:$

$V_{n}\equiv\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\left|V_{ij}(k)-E_{H_{0}}\left(V_{ij}(k)\right)\right|$

Note that we can allow $K$ to grow as $n$ increase. Let $N=Kp^{2}$. We define
the following assumption for any matrix ${\boldsymbol{\Sigma}}$:

  * (A1)

Let $\boldsymbol{\Sigma}=\left(\sigma_{ij}\right)_{1\leq i,j\leq N}$. For some
$\varrho\in(0,1),$ assume $\left|\sigma_{ij}\right|\leq\varrho$ for all $1\leq
i<j\leq N$ and $N\geq 2.$ Suppose $\left\\{\delta_{N};N\geq 1\right\\}$ and
$\left\\{\varsigma_{N};N\geq 1\right\\}$ are positive constants with
$\delta_{N}=o(1/\log N)$ and $\varsigma=\varsigma_{N}\rightarrow 0$ as
$N\rightarrow\infty.$ For $1\leq i\leq N,$ define $B_{N,i}=\left\\{1\leq j\leq
N;\left|\sigma_{ij}\right|\geq\delta_{N}\right\\}$ and $C_{N}=\left\\{1\leq
i\leq N;\left|B_{N,i}\right|\geq N^{\varsigma}\right\\}.$ We assume that
$\left|C_{N}\right|/N\rightarrow 0$ as $N\rightarrow\infty$.

Here we define
$\\{\nu_{1},\cdots,\nu_{Kp^{2}}\\}=\\{V_{ij}(k)/\sigma_{V}\\}_{1\leq i,j\leq
p,1\leq k\leq K}$. Define $\sigma^{V}_{ij}=\mathrm{cor}(\nu_{i},\nu_{j})$ and
${\boldsymbol{\Sigma}}_{V}=(\sigma_{ij}^{V})_{1\leq i,j\leq N}$.

To derive the limiting null distribution of simple linear rank statistics, we
need the following conditions.

  * (C1)

The regression constants $\left\\{c_{n-k,1},\ldots,c_{n-k,n-k}\right\\}$
satisfying

$\displaystyle\max_{1\leqslant i\leqslant
n-k}\left|c_{n-k,i}-\bar{c}_{n-k}\right|^{2}\leqslant\frac{C_{1}^{2}}{n-k}\sum_{i=1}^{n-k}\left(c_{n-k,i}-\bar{c}_{n-k}\right)^{2},$
$\displaystyle\left|\sum_{i=1}^{n-k}\left(c_{n-k,i}-\bar{c}_{n-k}\right)^{3}\right|^{2}\leqslant\frac{C_{2}^{2}}{n-k}\left\\{\sum_{i=1}^{n-k}\left(c_{n-k,i}-\bar{c}_{n-k}\right)^{2}\right\\}^{3}$

where $\bar{c}_{n-k}\equiv\sum_{i=1}^{n-k}c_{n-k,i}$ represents the sample
mean of the regression constants and $C_{1}$ and $C_{2}$ are two constants.

  * (C2)

The score function $g(\cdot)$ is differentiable with bounded Lipschitz
constant.

  * (C3)

The correlation matrix ${\boldsymbol{\Sigma}}_{V}$ satisfies Assumption (A1).

Remark: The assumption (A1) is the same as the condition (2.2) in Feng et al.
(2022a), which demands the number of variables that are strongly-correlated
with many other variables should not be too much. If the eigenvalues of
${\boldsymbol{\Sigma}}_{V}$ are all bounded, we have $\max_{1\leq i\leq
N}\sum_{j=1}^{N}\sigma_{ij}^{V2}\leq C$ for some constant $C>0$. Then, let
$\delta_{N}=(\log N)^{-2}$ for $N\geq e^{e}$, so for each $1\leqslant
i\leqslant
N,\delta_{N}^{2}\cdot\left|B_{N,i}\right|\leqslant\sum_{j=1}^{N}\sigma_{ij}^{V2}\leqslant
C$. Hence, $\left|B_{N,i}\right|\leqslant C\cdot(\log N)^{2}<N^{\kappa}$ where
$\kappa=\kappa_{N}:=5(\log\log N)/\log N$ for large $N$ . As a result,
$\left|C_{N}\right|=0$ and condition (C3) holds. Condition (C1) is commonly
used to deviate the asymptotical normality of the simple linear rank
statistics, see Hájek et al. (1999) and Kallenberg (1982). If $f$ is a linear
function, Condition (C1) will hold directly.

Next, we state the theoretical result about the limiting null distribution of
simple linear rank statistics.

######  Theorem 1.

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

$\left|\operatorname{P}\left(V_{n}^{2}/\sigma_{V}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leqslant
y\right)-\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}\right|=o(1)$

where $\sigma_{V}^{2}=\mathrm{var}_{H_{0}}(V_{ij}(k))$ if $N=o(n^{\epsilon})$
as $n\to\infty$ for some positive constant $\epsilon$.

We propose the following size-$\alpha$ test $T^{V}_{\alpha}$ of $H_{0}$:

$T_{\alpha}^{V}\doteq
I\left(V_{n}^{2}/\sigma_{V}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right),$ (2.4)

where $q_{\alpha}=-\log(\pi)-2\log\log(1-\alpha)^{-1}$.

To specify the alternative hypothesis, we introduce a notation for a set of
vectors which satisfying some specific condition. Define $N=Kp^{2}$. Let
$\mathcal{U}(c)$ be a set of vectors indexed by a constant $c$:

$\mathcal{U}(c)\equiv\left\\{M=(m_{l})_{1\leq l\leq
N}\in\mathbb{R}^{N}\mid\max_{1\leq l\leq N}m_{l}\geqslant c(\log
N)^{1/2}\right\\}.$

Based on the above definition, we know that $\mathcal{U}(c)$ is the set of
vectors of which at least one element has magnitude greater than $c(\log
N)^{1/2}$ for some large enough constant $c>0$.

Next, we specify the sparse local alternative based on $\mathcal{U}(c)$. We
define the random vector
$\hat{V}=\left[\hat{V}_{ij}(k)\right]\in\mathbb{R}^{N}$ by

$\hat{V}_{ij}(k)=\sigma_{V}^{-1}\left\\{V_{ij}(k)-E_{H_{0}}\left(V_{ij}(k)\right)\right\\},\quad(1\leqslant
i,j\leqslant p;1\leqslant k\leqslant K)$

where $\sigma_{V}$ is defined in (2.3) and $\left\\{V_{ij}(k),1\leqslant
i,j\leqslant p,1\leq k\leq K\right\\}$ are the simple linear rank statistics.
Let the population version of $\hat{V}$ be $V\equiv E(\hat{V})$. We study the
power of tests against the alternative

$H_{\mathrm{a}}^{V}(c)\equiv\\{F(\boldsymbol{\varepsilon}):V\\{F(\boldsymbol{\varepsilon})\\}\in\mathcal{U}(c)\\}$

where $F(\boldsymbol{\varepsilon})$ is the joint distribution function of
$\boldsymbol{\varepsilon}$ and we write $V\\{F(\boldsymbol{\varepsilon})\\}$
to emphasize that
$V=E(\hat{V})=\int\hat{V}\mathrm{d}F(\boldsymbol{\varepsilon})$ is a function
of $F(\boldsymbol{\varepsilon})$.

The following theorem now describe the conditions under which the power of the
test based on simple linear rank statistics converges to one as $n$ and $p$
going to infinity, under the sparse local alternative $H_{\mathrm{a}}^{V}$.

######  Theorem 2.

Assume Conditions (C1)-(C3) hold. And assume that
$\sigma_{V}^{2}=A_{1}\\{1+o(1)\\}$ and $\max\\{|f(0)|,|g(0)|\\}\leqslant
A_{2}$ for some positive constants $A_{1}$ and $A_{2}$. Further assume that
$f(\cdot)$ and $g(\cdot)$ have bounded Lipschitz constants. Then, for some
large scalar $B_{1}$ depending only on $A_{1},A_{2}$ and the Lipschitz
constants of $f(\cdot)$ and $g(\cdot)$

$\inf_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{V}\left(B_{1}\right)}\operatorname{pr}\left(T^{V}_{\alpha}=1\right)=1-o(1)$

where the infimum is taken over all distributions
$F(\boldsymbol{\varepsilon})$ such that
$V\\{F(\boldsymbol{\varepsilon})\\}\in\mathcal{U}\left(B_{1}\right)$.

Define $r_{ij}(k)$ is the correlation between $\varepsilon_{t,i}$ and
$\varepsilon_{t+k,j}$. To investigate the rate optimality of the test based on
simple linear rank statistics, we need the following assumption for the
distribution:

  * (A2)

When $\boldsymbol{\varepsilon}$ is Gaussian, suppose that for large $n$ and
$p$, $cV_{ij}(k)\leqslant r_{ij}(k)\leqslant CV_{ij}(k)$ for $1\leq i,j\leq
p,1\leq k\leq K$ with probability tending to one, where $c$ and $C$ are two
constants.

For each $n$, define $\mathcal{T}_{\alpha}$ to be the set of all measurable
size-$\alpha$ tests. In other words,
$\mathcal{T}_{\alpha}:=\\{T_{\alpha}:\operatorname{pr}(T_{\alpha}=1|H_{0})\leq\alpha\\}$.

Finally, the rate optimality result can be stated by the following theorem.
Recall that $T_{\alpha}^{V}$ defined in (2.4) can correctly reject the null
hypothesis provided that at least one element in $V$ has magnitude greater
than $c(\log N)^{1/2}$ for some constant $c$. In the following theorem, we
show that the rate of the signal gap $(\log N)^{1/2}$ cannot be further
relaxed.

######  Theorem 3.

Suppose that the simple linear rank statistics $\left\\{V_{ij}(k),1\leqslant
i,j\leqslant p,1\leq k\leq K\right\\}$ satisfy all the conditions in Theorems
1 and 2. Suppose also that Assumption (A2) holds. Then, the corresponding
size-$\alpha$ test $T^{V}_{\alpha}$ is rate-optimal. In other words, there
exist two constants $D_{1}<D_{2}$ such that:

  * (i)

$\sup_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{V}\left(D_{2}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)=o(1)$;

  * (ii)

for any $\beta>0$ satisfying $\alpha+\beta<1,$ for large $n$ and $p$ we have

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{V}\left(D_{1}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)\geqslant
1-\alpha-\beta$

The above theorem means that any measurable size-$\alpha$ test cannot
distinguish between the null hypothesis and the sparse alternative when the
coefficient $c$ in $H_{\mathrm{a}}^{V}(c)$ is small enough.

As an example of simple linear rank statistic, we state the high-dimensional
white noise test based on the Spearman’s rho as follows.

##### Example 1 (Spearman’s rho).

Recall that $Q_{n-k,t}^{i}(k)$ and $\tilde{Q}_{n-k,t+k}^{j}(k)$ be the ranks
of $\varepsilon_{t,i}$ and $\varepsilon_{t+k,j}$ among
$\left\\{\varepsilon_{1,i},\ldots,\varepsilon_{n-k,i}\right\\}$ and
$\\{\varepsilon_{k+1,j},\cdots,\varepsilon_{n,j}\\}$, respectively. Let
$R_{n-k,t+k}^{ij}(k)$ be the relative rank of $\varepsilon_{t+k,j}$ compared
to $\varepsilon_{t,i}$; that is,
$R_{n-k,t+k}^{ij}(k)\equiv\tilde{Q}_{n-k,t^{\prime}+k}^{j}(k)$ subject to the
constraint that $Q_{n-k,t^{\prime}}^{i}(k)=t$ for $t=1,\cdots,n-k$. Spearman’s
rho is defined as

$\displaystyle\rho_{ij}(k)$
$\displaystyle=\frac{\sum_{t=1}^{n-k}\left(Q_{n-k,t}^{i}(k)-\bar{Q}_{n-k}^{i}(k)\right)\left(\tilde{Q}_{n-k,t+k}^{j}(k)-\bar{\tilde{Q}}_{n-k}^{j}(k)\right)}{\left\\{\sum_{t=1}^{n-k}\left(Q_{n-k,t}^{i}(k)-\bar{Q}_{n-k}^{i}(k)\right)^{2}\sum_{t=1}^{n-k}\left(\tilde{Q}_{n-k,t+k}^{j}(k)-\bar{\tilde{Q}}_{n-k}^{j}(k)\right)^{2}\right\\}^{1/2}}$
$\displaystyle=\frac{12}{(n-k)\left((n-k)^{2}-1\right)}\sum_{t=1}^{n-k}\left(i-\frac{n-k+1}{2}\right)\left(R_{n-k,t+k}^{ij}(k)-\frac{n-k+1}{2}\right)\quad(i,j\in\\{1,\ldots,p\\})$

where $\bar{Q}_{n-k}^{i}(k)=\bar{\tilde{Q}}_{n-k}^{j}(k)\equiv(n-k+1)/2.$ This
is a simple linear rank statistic, and we have

$E_{H_{0}}\left(\rho_{ij}(k)\right)=0,\quad\operatorname{var}_{H_{0}}\left(\rho_{ij}(k)\right)=(n-k-1)^{-1}\quad(i,j\in\\{1,\ldots,p\\})$

According to (2.4), the corresponding test statistic is

$L_{\rho}=I\left\\{\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}(n-k)\rho_{ij}(k)^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geqslant
q_{\alpha}\right\\}$

where $q_{\alpha}\equiv-\log(\pi)-2\log\log(1-\alpha)^{-1}$.

###  2.2 Non-degenerate U-statistics

The second family includes the tests based on non-degenerate U-statistics of
the form (Han et al. 2017)

$\displaystyle U_{n}=\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}(n-k)^{1/2}\left|{U}_{ij}(k)-E_{H_{0}}(U_{ij}(k))\right|$ (2.5)

where $A_{n-k}^{m}=(n-k)(n-k-1)\cdots(n-k-m+1)$,

$\displaystyle{U}_{ij}(k)=$ $\displaystyle\frac{1}{A_{n-k}^{m}}\sum_{1\leq
t_{1}\not=t_{2},\cdots,\not=t_{m}\leq
n-k}h((\varepsilon_{t_{1},i},\varepsilon_{t_{1}+k,j})^{\top},\cdots,(\varepsilon_{t_{m},i},\varepsilon_{t_{m}+k,j})^{\top})$
(2.6)

Here $U_{ij}(k)$ depends only on $\\{R_{n-k,t}^{ij}(k)\\}_{t=k+1}^{n}$. For
our purposes $h$ may always be assumed to be bounded but not necessarily
symmetric. The boundedness assumption is mild since correlation is the object
of interest.

Further concepts concerning U-statistics are needed to state the assumption
for the derivation of the limiting null distribution. For
$m\in\mathbb{Z}^{+}$, we define $[m]=\\{1,2,\cdots,m\\}$ and write
$\mathcal{P}_{m}$ for the set of all $m!$ permutations of $[m]$. For any
kernel $h(\cdot),$ any number $\ell\in[m],$ and any measure
$\mathbb{P}_{\boldsymbol{Z}},$ we write

$h_{\ell}\left(\boldsymbol{z}_{1}\ldots,\boldsymbol{z}_{\ell};\mathbb{P}_{\boldsymbol{Z}}\right):=\mathbb{E}h\left(\boldsymbol{z}_{1}\ldots,\boldsymbol{z}_{\ell},\boldsymbol{Z}_{\ell+1},\ldots,\boldsymbol{Z}_{m}\right)$
(2.7)

and

$h^{(\ell)}\left(\boldsymbol{z}_{1},\ldots,\boldsymbol{z}_{\ell};\mathbb{P}_{\boldsymbol{Z}}\right):=h_{\ell}\left(\boldsymbol{z}_{1},\ldots,\boldsymbol{z}_{\ell};\mathbb{P}_{\boldsymbol{Z}}\right)-\mathbb{E}h-\sum_{k=1}^{\ell-1}\sum_{1\leq
i_{1}<\cdots<i_{k}\leq\ell}h^{(k)}\left(\boldsymbol{z}_{i_{1}},\ldots,\boldsymbol{z}_{i_{k}};\mathbb{P}_{\boldsymbol{Z}}\right)$
(2.8)

where $\boldsymbol{Z}_{1},\ldots,\boldsymbol{Z}_{m}$ are $m$ independent
random vectors with distribution $\mathbb{P}_{\boldsymbol{Z}}$ and
$\mathbb{E}h:=\mathbb{E}h\left(\boldsymbol{Z}_{1},\ldots,\boldsymbol{Z}_{m}\right)$.
The kernel as well as the corresponding U-statistic is non-degenerate under
$\mathbb{P}_{\boldsymbol{Z}}$ if the variance of $h_{1}(\cdot)$ is not zero.

Based on above definitions, we state the following conditions which are needed
to derive the limiting null distribution.

  * (C4)

The kernel function $h(\cdot)$ is bounded and non-degenerate.

  * (C5)

The correlation matrix of $U_{ij}(k)$–${\boldsymbol{\Sigma}}_{U}$ satisfies
Assumption (A1).

The following theorem show the asymptotic distribution of the non-degenerate
U-statistics under the null hypothesis.

######  Theorem 4.

Suppose (C4)-(C5) hold. Then under $H_{0}$, for any $y\in\mathbb{R}$ we have

$\left|\operatorname{P}\left(U_{n}^{2}/\sigma_{U}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leqslant
y\right)-\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}\right|=o(1)$

where $\sigma_{U}^{2}=(n-k)\mathrm{var}_{H_{0}}(U_{ij}(k))$ if
$N=o(n^{\epsilon})$ as $n\to\infty$ for some positive constant $\epsilon$.

We propose the following size-$\alpha$ test $T^{U}_{\alpha}$ of $H_{0}$:

$T_{\alpha}^{U}\doteq
I\left(U_{n}^{2}/\sigma_{U}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right)$ (2.9)

where $q_{\alpha}=-\log(\pi)-2\log\log(1-\alpha)^{-1}$.

To specify the sparse local alternative for the tests based on non-degenerate
U-statistics, we first define the random vector
$\hat{U}=\left[\hat{U}_{ij}(k)\right]\in\mathbb{R}^{N}$ by

$\hat{U}_{ij}(k)=\sigma_{U}^{-1}(n-k)^{1/2}\left\\{U_{ij}(k)-E_{H_{0}}\left(U_{ij}(k)\right)\right\\},\quad(1\leqslant
i,j\leqslant p;1\leqslant k\leqslant K)$

where $\sigma_{U}$ is defined in Theorem 4 and $\left\\{U_{ij}(k),1\leqslant
i,j\leqslant p,1\leq k\leq K\right\\}$ are the non-degenerate U-statistics.
Let the population version of $\hat{U}$ be $U\equiv E(\hat{U}).$ We study the
power of tests against the alternative

$H_{\mathrm{a}}^{U}(c)\equiv\\{F(\boldsymbol{\varepsilon}):U\\{F(\boldsymbol{\varepsilon})\\}\in\mathcal{U}(c)\\}$

where $F(\boldsymbol{\varepsilon})$ is the joint distribution function of
$\boldsymbol{\varepsilon}$ and we write $U\\{F(\boldsymbol{\varepsilon})\\}$
to emphasize that
$U=E(\hat{U})=\int\hat{U}\mathrm{d}F(\boldsymbol{\varepsilon})$ is a function
of $F(\boldsymbol{\varepsilon})$.

The following theorem states the conditions which are required to establish
the convergence of the power of ${T}^{U}_{\alpha}$ to one as $n$ and $p$ going
to infinity under the sparse alternative.

######  Theorem 5.

Suppose that the kernel function $h(\cdot)$ in (2.6) is bounded with
$|h(\cdot)|\leqslant A_{3}$ and

$m^{2}\operatorname{var}_{H_{0}}\left[E_{H_{0}}\left\\{h\left((X_{11},X_{12})^{\top},\ldots,(X_{m1},X_{m2})^{\top}\right)\mid(X_{11},X_{12})^{\top}\right\\}\right]=A_{4}\\{1+o(1)\\}$

for some positive constants $A_{3}$ and $A_{4}.$ Then, for some large scalar
$B_{2}$ depending only on $A_{3},A_{4}$ and $m$,

$\inf_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{U}\left(B_{2}\right)}P\left({T}_{\alpha}=1\right)=1-o(1)$

where the infimum is taken over all distributions
$F(\boldsymbol{\varepsilon})$ such that
$U\\{F(\boldsymbol{\varepsilon})\\}\in\mathcal{U}\left(B_{2}\right).$

To study the rate optimality in terms of power for the tests based on non-
degenerate U-statistics, we need the following assumption for the
distribution:

  * (A3)

When $\boldsymbol{\varepsilon}$ is Gaussian, suppose that for non-degenerate
U-statistics $U_{ij}(k)$ and large $n$ and $p$, $cU_{ij}(k)\leqslant
r_{ij}(k)\leqslant CU_{ij}(k)$ for $1\leq i,j\leq p,1\leq k\leq K$ with
probability tending to one, where $c$ and $C$ are two constants.

The rate optimality result and related conditions for the tests based on the
non-degenerate U-statistics can be shown as follows, which implies that the
rate of the signal gap $(\log N)^{1/2}$ cannot be further relaxed.

######  Theorem 6.

Suppose that Non-degenerate U-statistics $\left\\{U_{ij}(k),1\leqslant
i,j\leqslant p,1\leq k\leq K\right\\}$ satisfy all the conditions in Theorems
4 and 5. Suppose also that Assumption (A3) holds. Then, the corresponding
size-$\alpha$ test $T^{U}_{\alpha}$ is rate-optimal. In other words, there
exist two constants $D_{3}<D_{4}$ such that:

  * (i)

$\sup_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{U}\left(D_{4}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)=o(1)$;

  * (ii)

for any $\beta>0$ satisfying $\alpha+\beta<1,$ for large $n$ and $p$ we have

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{F(\boldsymbol{\varepsilon})\in
H_{\mathrm{a}}^{U}\left(D_{3}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)\geqslant
1-\alpha-\beta.$

As an example of non-degenerate U-statistics, we state the high-dimensional
white noise test based on the Kendall’s tau as follows.

##### Example 2 (Kendall’s tau).

Kendall’s tau is defined, for $i,j\in\\{1,\ldots,p\\},$ by

$\displaystyle\tau_{ij}(k)$ $\displaystyle=\frac{2}{(n-k)(n-k-1)}\sum_{1\leq
l<l^{\prime}\leq
n-k}\operatorname{sign}\left(\varepsilon_{l^{\prime},i}-\varepsilon_{l,i}\right)\operatorname{sign}\left(\varepsilon_{l^{\prime}+k,j}-\varepsilon_{l+k,j}\right)$
$\displaystyle=\frac{2}{(n-k)(n-k-1)}\sum_{1\leq l<l^{\prime}\leq
n-k}\operatorname{sign}\left(R_{n-k,l^{\prime}+k}^{ij}(k)-R_{n-k,l+k}^{ij}(k)\right)$

where the sign function $\operatorname{sign}(\cdot)$ is defined as
$\operatorname{sign}(x)=x/|x|$ with the convention $0/0=0$. This statistic is
a function of the relative ranks
$\left\\{R_{n-k,t+k}^{ij}(k),t=1,\ldots,n-k\right\\}$ and is also a
$U$-statistic with bounded kernel
$h\left(x_{1,\\{1,2\\}},x_{2,\\{1,2\\}}\right)\equiv\operatorname{sign}\left(x_{1,1}-x_{2,1}\right)\operatorname{sign}\left(x_{1,2}-x_{2,2}\right)$.
Accordingly, Kendall’s tau is a rank-type $U$-statistic. Moreover,

$E_{H_{0}}\left(\tau_{ij}(k)\right)=0,\quad\operatorname{var}_{H_{0}}\left(\tau_{ij}(k)\right)=\frac{2(2(n-k)+5)}{9(n-k)(n-k-1)}\quad(i,j\in\\{1,\ldots,p\\})$

According to (8), the proposed test statistic based on Kendall’s tau is

$L_{\tau}=I\left\\{\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\frac{9(n-k)(n-k-1)}{2(2(n-k)+5)}\tau_{ij}(k)^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geqslant
q_{\alpha}\right\\}$

###  2.3 Degenerate U-statistics

The third family includes the tests based on degenerate U-statistics, which
are very useful to detect the non-linear and non-monotone relationships in the
autocorrelations. We use the term completely degenerate to indicate that the
variances of $h_{1}(\cdot),\ldots,h_{m-1}(\cdot)$ are all zero. Finally, let
$\mathbb{P}_{0}$ be the uniform distribution on $[0,1],$ and write
$\mathbb{P}_{0}\otimes\mathbb{P}_{0}$ for its product measure, the uniform
distribution on $[0,1]^{2}$.

In order to derive the limiting null distribution and establish the
theoretical results related to the power of the tests based on the degenerate
U-statistics, we need the following assumption concerning the kernel function
$h$.

  * (C6)

The kernel $h$ is rank-based, symmetric, and has the following three
properties:

    * (i)

$h$ is bounded.

    * (ii)

$h$ is mean-zero and degenerate under independent continuous margins, i.e.,
$\mathbb{E}\left\\{h_{1}\left(\boldsymbol{Z}_{1};\mathbb{P}_{0}\otimes\right.\right.$
$\left.\left.\mathbb{P}_{0}\right)\right\\}^{2}=0$ as
$\boldsymbol{Z}_{1}\sim\mathbb{P}_{0}\otimes\mathbb{P}_{0}$

    * (iii)

$h_{2}\left(\boldsymbol{z}_{1},\boldsymbol{z}_{2};\mathbb{P}_{0}\otimes\mathbb{P}_{0}\right)$
has uniformly bounded eigenfunctions, that is, it admits the expansion

$h_{2}\left(\boldsymbol{z}_{1},\boldsymbol{z}_{2};\mathbb{P}_{0}\otimes\mathbb{P}_{0}\right)=\sum_{v=1}^{\infty}\lambda_{v}\phi_{v}\left(\boldsymbol{z}_{1}\right)\phi_{v}\left(\boldsymbol{z}_{2}\right)$

where $\left\\{\lambda_{v}\right\\}$ and $\left\\{\phi_{v}\right\\}$ are the
eigenvalues and eigenfunctions satisfying the integral equation

$\displaystyle\mathbb{E}h_{2}\left(\boldsymbol{z}_{1},\boldsymbol{Z}_{2}\right)\phi\left(\boldsymbol{Z}_{2}\right)$
$\displaystyle=\lambda\phi\left(\boldsymbol{z}_{1}\right)\text{ for all
}\boldsymbol{z}_{1}\in\mathbb{R}^{2}$ $\displaystyle\text{with
}\boldsymbol{Z}_{2}\sim\mathbb{P}_{0}\otimes\mathbb{P}_{0},\lambda_{1}\geq\lambda_{2}\geq\cdots\geq
0,\Lambda$ $\displaystyle=\sum_{v=1}^{\infty}\lambda_{v}\in(0,\infty),\text{
and }\sup_{v}\left\|\phi_{v}\right\|_{\infty}<\infty.$

The first requirement about boundedness property can be easily verified for
the rank correlations which are commonly used, for example, Spearman’s rho,
Kendall’s tau and many others. The other two requirements are much more
specific, but can be satisfied by some typical rank correlation measures as
long as their consistency properties are known. Moreover, it is easy to see
that the assumption $\Lambda>0$ implies $\lambda_{1}>0,$ so that
$h_{2}(\cdot)$ is not a constant function.

We also need the following condition to derive the limiting null distribution
for the degenerate U-statistics. We first make several definitions which will
be used in the following condition. Define a quantity $\theta$, which is any
absolute constant such that

$\displaystyle\theta<\sup\left\\{q\in[0,1/3):\sum_{v>[n^{(1-3q)/5}]}\lambda_{v}=O(n^{-q})\right\\}$

if infinitely many eigenvalues $\lambda_{v}$ are nonzero, and $\theta=1/3$
otherwise. Define
$\omega_{l,v}=(n-k_{l})^{-1/2}\sum_{t=1}^{n-k_{l}}\phi_{v}(Z_{t,l})$ for
$l=1,\cdots,N$, $v=1,\cdots,M$ where $M=[n^{(1-3\theta)/5}]$ and $Z_{t,l}$ is
the corresponding $Z$ of $U_{ij}(k)$ in Condition (C6). Let
$b_{lv,rs}=\mathrm{cov}(\omega_{l,v},\omega_{r,s})$ for $1\leq l,r\leq N,1\leq
v,s\leq M$. Let $\boldsymbol{\omega}_{l}=(\omega_{l,1},\cdots,\omega_{l,M})$
and $\Xi_{l}={\boldsymbol{\Sigma}}_{l}{\boldsymbol{\Sigma}}_{l}^{\top}$ where
${\boldsymbol{\Sigma}}_{l}\in\mathbb{R}^{M\times(N-1)M}$ is the covariance
matrix between $\boldsymbol{\omega}_{l}$ with
$\boldsymbol{\omega}_{r},r\in\\{1,\cdots,N\\}\setminus\\{l\\}$.

  * (C7)

There exists a constant $\delta\in(0,1)$ satisfying
$\lambda_{max}(\Xi_{l})\leq\delta$ for all $1\leq l\leq N$. Suppose
$\left\\{\delta_{N};N\geq 1\right\\}$ and $\left\\{\varsigma_{N};N\geq
1\right\\}$ are positive constants with $\delta_{N}=o(1/\log N)$ and
$\varsigma=\varsigma_{N}\rightarrow 0$ as $N\rightarrow\infty.$ Let
$\Xi_{ij}=\mathrm{cov}(\boldsymbol{\omega}_{i},\boldsymbol{\omega}_{j})$. For
$1\leq i\leq N,$ define $B_{N,i}=\left\\{1\leq j\leq
N\mid\lambda_{max}(\Xi_{ij}\Xi_{ij}^{\top})\geq\delta_{N}^{2+2c}\right\\}$ for
some constant $c>0$ and $C_{N}=\left\\{1\leq i\leq N;\left|B_{N,i}\right|\geq
N^{\varsigma}\right\\}.$ We assume that $\left|C_{N}\right|/N\rightarrow 0$ as
$N\rightarrow\infty$.

In the following theorem, we show the limiting null distribution and related
conditions for the degenerate U-statistics.

######  Theorem 7.

Under conditions (C6)-(C7). Then for any absolute constant $y\in\mathbb{R}$
that

$\displaystyle\mathbb{P}\left\\{\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\frac{n-k-1}{\lambda_{1}\left(\begin{array}[]{c}m\\\
2\end{array}\right)}{U}_{ij}(k)-2\log(Kp^{2})-\left(\mu_{1}-2\right)\log\log(Kp^{2})+\frac{\Lambda}{\lambda_{1}}\leq
y\right\\}$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\exp\left(-\frac{y}{2}\right)\right\\}+o(1)$

for $\log N=o(n^{\theta})$ as $n\to\infty$. Here $\mu_{1}$ is the multiplicity
of the largest eigenvalue $\lambda_{1}$ in the sequence
$\\{\lambda_{1},\lambda_{2},\ldots\\}$,
$\kappa:=\prod_{v=\mu_{1}+1}^{\infty}(1-\lambda_{v}/\lambda_{1})^{-1/2}$ and
$\Gamma(z):=\int_{0}^{\infty}x^{z-1}e^{-x}dx$ is the gamma function.

We propose the following size-$\alpha$ test $T_{\alpha}^{D}$ for degenerate
U-statistics:

$T_{\alpha}^{D}=I\left(\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\frac{n-k-1}{\lambda_{1}\left(\begin{array}[]{c}m\\\
2\end{array}\right)}{U}_{ij}(k)-2\log(Kp^{2})-\left(\mu_{1}-2\right)\log\log(Kp^{2})+\frac{\Lambda}{\lambda_{1}}\geq\tilde{q}_{\alpha}\right)$
(2.10)

where $\tilde{q}_{\alpha}$ is the $1-\alpha$ quantile of the Gumbel
distribution function
$\exp\left\\{-{\kappa}/{\Gamma\left(\mu_{1}/2\right)}\exp\left(-{y}/{2}\right)\right\\}$,
i.e.,

$\tilde{q}_{\alpha}=-\log\left(\frac{\Gamma^{2}\left(\mu_{1}/2\right)}{\kappa^{2}}\right)-2\log\log(1-\alpha)^{-1}.$

It is easy to show that
$\mathbb{P}_{H_{0}}\left(T_{\alpha}^{D}=1\right)=\alpha+o(1)$

We study the power of the proposed test based on the degenerate U-statistics
from now on. It is necessary to introduce a new distribution family which is
also useful to specify the alternative. Recall the definition of
$h^{(1)}(\cdot)$ in (2.8) . For any kernel function $h(\cdot)$ and constants
$\gamma>0$ and $n,p\in\mathbb{Z}^{+},$ define a general $p$-dimensional (not
necessarily continuous) distribution family as follows:

$\mathcal{D}(\gamma,np;h):=\left\\{F(\boldsymbol{X}):\boldsymbol{X}\in\mathbb{R}^{np},\operatorname{Var}_{ijk}\left\\{h^{(1)}\left(\cdot;\mathbb{P}_{ijk}\right)\right\\}\leq\gamma\mathbb{E}_{ijk}h\text{
for all }1\leq i,j\leq p,1\leq k\leq K\right\\}$

where $F(\boldsymbol{X})$ is the distribution (law) of $\boldsymbol{X},$ and
$\mathbb{P}_{ijk},\mathbb{E}_{ijk}(\cdot),$ and
$\operatorname{Var}_{ijk}(\cdot)$ stand for the probability measure,
expectation, and variance operated on the bivariate distribution of
$\left(\varepsilon_{ti},\varepsilon_{t+k,j}\right)^{\top},$ respectively. The
family $\mathcal{D}(\gamma,np;h)$ intrinsically characterizes the slope of the
non-negative function
$\operatorname{Var}_{ijk}\left\\{h^{(1)}\left(\cdot;\mathbb{P}_{ijk}\right)\right\\}$
with regard to the dependence between $\varepsilon_{ti}$ and
$\varepsilon_{t+k,j},$ characterized by the non-negative correlation measure
$\mathbb{E}_{ijk}h$. Under the null hypothesis, we have

$\operatorname{Var}_{ijk}\left\\{h^{(1)}\left(\cdot;\mathbb{P}_{ijk}\right)\right\\}=\mathbb{E}_{ijk}h=0$

provided that Condition (C6) holds for $h(\cdot)$. Therefore, as the
dependence between $\varepsilon_{ti}$ and $\varepsilon_{t+k,j}$ increasing, it
can be expected that the variance
$\operatorname{Var}_{ijk}\left\\{h^{(1)}\left(\cdot;\mathbb{P}_{ijk}\right)\right\\}$
will depart away from zero with the same or a slower rate compared to
$\mathbb{E}_{ijk}h$.

In the following theorem, we show that the power of the proposed test
$T^{D}_{\alpha}$ converges to one as $n$ and $p$ increasing to infinity under
a newly specified sparse alternative.

######  Theorem 8.

Given any $\gamma>0$ and a kernel $h(\cdot)$ satisfying Condition (C6), there
exists some sufficiently large $B_{3}$ depending on $\gamma$ such that

$\inf_{F(\boldsymbol{\varepsilon})\in\mathcal{D}(\gamma,np;h)\cap
H_{\mathrm{a}}^{U}\left(B_{3}\right)}\mathbb{P}\left(T^{D}_{\alpha}=1\right)=1-o(1)$

The establishment of the rate optimality of the tests based on degenerate
U-statistics requires the following assumption for the distribution:

  * (A4)

When $\boldsymbol{\varepsilon}$ is Gaussian, suppose that for degenerate
U-statistics $U_{ij}(k)$ and large $n$ and $p$, $cU_{ij}(k)\leqslant
r_{ij}(k)\leqslant CU_{ij}(k)$ for $1\leq i,j\leq p,1\leq k\leq K$ with
probability tending to one, where $c$ and $C$ are two constants.

Under the new type of sparse local alternative, we could show the rate
optimality in terms of power for the proposed test in the following theorem.

######  Theorem 9.

Suppose that Degenerate U-statistics $\left\\{U_{ij}(k),1\leqslant
i,j\leqslant p,1\leq k\leq K\right\\}$ satisfy all the conditions in Theorems
7 and 8. Suppose also that Assumption (A4) holds. Then, the corresponding
size-$\alpha$ test $T^{D}_{\alpha}$ is rate-optimal. In other words, there
exist two constants $D_{5}<D_{6}$ such that:

  * (i)

$\sup_{F(\boldsymbol{\varepsilon})\in\mathcal{D}(\gamma,np;h)\cap
H_{\mathrm{a}}^{U}\left(D_{6}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)=o(1)$;

  * (ii)

for any $\beta>0$ satisfying $\alpha+\beta<1,$ for large $n$ and $p$ we have

$\inf_{T_{\alpha}\in\mathcal{T}_{\alpha}}\sup_{F(\boldsymbol{\varepsilon})\in\mathcal{D}(\gamma,np;h)\cap
H_{\mathrm{a}}^{U}\left(D_{5}\right)}\operatorname{pr}\left(T_{\alpha}=0\right)\geqslant
1-\alpha-\beta$

Three examples belonging to the family of degenerate U-statistics are provided
to test the high dimensional white noise as follows.

##### Example 3 (Hoeffding’s $D$).

The Hoeffding’s $D$ statistic is a rank-based U-statistic of order 5, which is
based on the symmetric kernel

$\begin{array}[]{l}h_{D}\left(z_{1},\ldots,z_{5}\right):=\frac{1}{16}\sum_{\left(i_{1},\ldots,i_{5}\right)\in\mathcal{P}_{5}}\\\
{\left[\left\\{I\left(z_{i_{1},1}\leq
z_{i_{5},1}\right)-I\left(z_{i_{2},1}\leq
z_{i_{5},1}\right)\right\\}\left\\{I\left(z_{i_{3},1}\leq
z_{i_{5},1}\right)-I\left(z_{i_{4},1}\leq
z_{i_{5},1}\right)\right\\}\right]}\\\ {\left[\left\\{I\left(z_{i_{1},2}\leq
z_{i_{5},2}\right)-I\left(z_{i_{2},2}\leq
z_{i_{5},2}\right)\right\\}\left\\{I\left(z_{i_{3},2}\leq
z_{i_{5},2}\right)-I\left(z_{i_{4},2}\leq
z_{i_{5},2}\right)\right\\}\right]}.\end{array}$

Thus, the Hoeffding’s $D$ correlation measure is given by $\mathbb{E}h_{D}$.
Based on Weihs et al. (2018, Proposition 7) or Nandy et al. (2016, Theorem
4.4), under the measure $\mathbb{P}_{0}\otimes\mathbb{P}_{0}$, the eigenvalues
and corresponding eigenfunctions of $h_{D,2}(\cdot)$ are:

$\lambda_{i,j;D}=3/\left(\pi^{4}i^{2}j^{2}\right)>0,\quad
i,j\in\mathbb{Z}^{+}$

and

$\phi_{i,j;D}\left\\{\left(z_{1,1},z_{1,2}\right)^{\top}\right\\}=2\cos\left(\pi
iz_{1,1}\right)\cos\left(\pi jz_{1,2}\right),\quad i,j\in\mathbb{Z}^{+},$

where $\Lambda_{D}:=\sum_{i,j}\lambda_{i,j;D}=1/12$ and
$\sup_{i,j}\left\|\phi_{i,j;D}\right\|_{\infty}\leq 2$. Therefore, by
considering the results in Hoeffding (1948), the kernel $h_{D}(\cdot)$
satisfies the three properties in Condition (C6). Based on the result in
Hoeffding (1948, p. 547), the correlation measure $\mathbb{E}h_{D}$ is non-
negative for arbitrary pair of random variables. Moreover, as shown by
Hoeffding (1948) and Yanagimoto (1970), for a pair of random variables which
is absolutely continuous in $\mathbb{R}^{2}$, the sufficient and necessary
condition for their independence is that $\mathbb{E}h_{D}=0$. However, this
result does not hold when the data is discrete or is continuous but not
absolute continuous, e.g. a counter example is given in Remark 1 of Yanagimoto
(1970).

Define
$\\{{\boldsymbol{X}}_{tijk}\\}=\\{(\varepsilon_{t,i},\varepsilon_{t+k,j})^{\top}\\}_{1\leq
t\leq n-k}$. According to (2.10), the corresponding test is

$\displaystyle\widehat{D}_{ij}(k):=$
$\displaystyle\left(\begin{array}[]{c}n-k\\\
5\end{array}\right)^{-1}\sum_{t_{1}<\cdots<t_{5}}h_{D}\left({\boldsymbol{X}}_{t_{1},ijk},\ldots,{\boldsymbol{X}}_{{t_{5}},ijk}\right)$
(2.13)

and

$\displaystyle L_{D}:=$ $\displaystyle I\left\\{\max_{1\leq k\leq
K}\max_{1\leq i,j\leq
p}\frac{\pi^{4}(n-k-1)}{30}\widehat{D}_{ij}(k)-2\log(Kp^{2})+\log\log(Kp^{2})+\frac{\pi^{4}}{36}>Q_{D,\alpha}\right\\}$
(2.14)

where
$Q_{D,\alpha}:=\log\left\\{\kappa_{D}^{2}/\pi\right\\}-2\log\log(1-\alpha)^{-1}$
and

$\kappa_{D}:=\left\\{2\prod_{n=2}^{\infty}\frac{\pi/n}{\sin(\pi/n)}\right\\}^{1/2}\approx
2.467.$

##### Example 4 (Blum-Kiefer-Rosenblatt’s $R$).

The Blum-Kiefer-Rosenblatt’s $R$ statistic (Blum et al. (1961)) is a rank-
based U-statistic of order 6, which is based on the symmetric kernel:

$\begin{array}[]{l}h_{R}\left(z_{1},\ldots,z_{6}\right):=\frac{1}{32}\sum_{\left(i_{1},\ldots,i_{6}\right)\in\mathcal{P}_{6}}\\\
{\left[\left\\{I\left(z_{i_{1},1}\leq
z_{i_{5},1}\right)-I\left(z_{i_{2},1}\leq
z_{i_{5},1}\right)\right\\}\left\\{I\left(z_{i_{3},1}\leq
z_{i_{5},1}\right)-I\left(z_{i_{4},1}\leq
z_{i_{5},1}\right)\right\\}\right]}\\\ {\left[\left\\{1\left(z_{i_{1},2}\leq
z_{i_{6},2}\right)-I\left(z_{i_{2},2}\leq
z_{i_{6},2}\right)\right\\}\left\\{I\left(z_{i_{3},2}\leq
z_{i_{6},2}\right)-1\left(z_{i_{4},2}\leq
z_{i_{6},2}\right)\right\\}\right]}.\end{array}$

The three properties in Condition (C6) can be easily verified based on the
fact that $h_{R,2}=2h_{D,2}$. Similarly, the correlation measure
$\mathbb{E}h_{R}$ is non-negative for arbitrary pair of random variables.
$\mathbb{E}h_{R}=0$ if and only if the pair of random variables are
independent (without requiring the continuity properties), see, e.g. page 490
of Blum et al. (1961).

According to (2.10), the corresponding test is

$\widehat{R}_{ij}(k):=\left(\begin{array}[]{c}n-k\\\
6\end{array}\right)^{-1}\sum_{t_{1}<\cdots<t_{6}}h_{R}\left({\boldsymbol{X}}_{t_{1},ijk},\ldots,{\boldsymbol{X}}_{t_{6},ijk}\right)$

and

$L_{R}:=I\left\\{\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\frac{\pi^{4}(n-k-1)}{90}\widehat{R}_{ij}(k)-2\log(Kp^{2})+\log\log(Kp^{2})+\frac{\pi^{4}}{36}>Q_{R,\alpha}\right\\}$

where $Q_{R,\alpha}:=Q_{D,\alpha}$

##### Example 5 (Bergsma-Dassios-Yanagimoto’s $\tau^{*}$).

Bergsma and Dassios (2014) introduced a rank correlation statistic as a
U-statistic of order 4 with the symmetric kernel

$\displaystyle h_{\tau^{*}}\left(z_{1},\ldots,z_{4}\right)$ $\displaystyle:=$
$\displaystyle\frac{1}{16}\sum_{\left(i_{1},\ldots,i_{4}\right)\in\mathcal{P}_{4}}\left\\{1\left(z_{i_{1},1},z_{i_{3},1}<z_{i_{2},1},z_{i_{4},1}\right)+I\left(z_{i_{2},1},z_{i_{4},1}<z_{i_{1},1},z_{i_{3},1}\right)\right.$
$\displaystyle\left.-I\left(z_{i_{1},1},z_{i_{4},1}<z_{i_{2},1},z_{i_{3},1}\right)-I\left(z_{i_{2},1},z_{i_{3},1}<z_{i_{1},1},z_{i_{4},1}\right)\right\\}$
$\displaystyle\left\\{I\left(z_{i_{1},2},z_{i_{3},2}<z_{i_{2},2},z_{i_{4},2}\right)+I\left(z_{i_{2},2},z_{i_{4},2}<z_{i_{1},2},z_{i_{3},2}\right)\right.$
$\displaystyle\left.-I\left(z_{i_{1},2},z_{i_{4},2}<z_{i_{2},2},z_{i_{3},2}\right)-I\left(z_{i_{2},2},z_{i_{3},2}<z_{i_{1},2},z_{i_{4},2}\right)\right\\},$

where
$I\left(y_{1},y_{2}<y_{3},y_{4}\right):=I\left(y_{1}<y_{3}\right)I\left(y_{1}<y_{4}\right)I\left(y_{2}<y_{3}\right)I\left(y_{2}<y_{4}\right)$.
Based on the fact that $h_{\tau^{*},2}=3h_{D,2}$, all properties in Condition
(C6) can be verified for $h_{\tau^{*}}(\cdot)$. As shown by Theorem 1 in
Bergsma and Dassios (2014), for a pair of random variables whose distribution
is discrete, absolutely continuous, or a mixture of both, the correlation
measure $\mathbb{E}h_{\tau^{*}}$ is non-negative and
$\mathbb{E}h_{\tau^{*}}=0$ if and only if the pair is independent.

According to (2.10), it yields the test

$\widehat{\tau}_{ij}^{*}(k):=\left(\begin{array}[]{c}n-k\\\
4\end{array}\right)^{-1}\sum_{t_{1}<\cdots<t_{4}}h_{\tau^{*}}\left({\boldsymbol{X}}_{t_{1},ijk},\ldots,{\boldsymbol{X}}_{t_{4},ijk}\right)$

and

$L_{\tau^{*}}:=I\left\\{\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\frac{\pi^{4}(n-k-1)}{54}\widehat{\tau}_{ij}^{*}(k)-2\log(Kp^{2})+\log\log(Kp^{2})+\frac{\pi^{4}}{36}>Q_{\tau^{*},\alpha}\right\\}$

where $Q_{\tau^{*},\alpha}:=Q_{D,\alpha}$

##  3 Simulation

In this section, we evaluate the empirical sizes and powers of several test
statistics based on Monte Carlo simulation. We mainly compare the performance
of the following test statistics:

  * •

$L_{r}$: the max-type test statistic provided by Chang et al. (2017);

  * •

$S_{r}$: the sum-type test statistic provided by Li et al. (2019);

  * •

$L_{\rho}$: the Spearman’s rho statistic defined in Example 1;

  * •

$L_{\tau}$: the Kendall’s tau statistic defined in Example 2;

  * •

$L_{D}$: the Hoeffding’s $D$ statistic defined in Example 3;

  * •

$L_{R}$: the Blum-Kiefer-Rosenblatt’s $R$ statistic defined in Example 4;

  * •

$L_{\tau^{*}}$: the Bergsma-Dassios-Yanagimoto’s $\tau^{*}$ statistic defined
Example 5.

###  3.1 Empirical sizes

Let $\boldsymbol{\varepsilon}_{t}={\bf A}{\boldsymbol{z}}_{t}$. We consider
the following four distribution for ${\boldsymbol{z}}_{t}$: (a)
${\boldsymbol{z}}_{t}\sim N(\boldsymbol{0},{\bf I}_{p})$; (b)
${\boldsymbol{z}}_{t}=\boldsymbol{w}_{t}^{1/3}$ with $\boldsymbol{w}_{t}\sim
N(\boldsymbol{0},{\bf I}_{p})$; (c)
${\boldsymbol{z}}_{t}=\boldsymbol{w}_{t}^{3}$ with $\boldsymbol{w}_{t}\sim
N(\boldsymbol{0},{\bf I}_{p})$; (d)
${\boldsymbol{z}}_{t}=(z_{t1},\cdots,z_{tp})^{\top}$ with
$z_{ti}\stackrel{{\scriptstyle i.i.d}}{{\sim}}t(3)/\sqrt{3}$. For the Models
(i)-(iv), we consider ${\bf A}={\boldsymbol{\Sigma}}^{1/2}$ with
${\boldsymbol{\Sigma}}=(0.5^{|i-j|})_{1\leq i,j\leq p}$ and
${\boldsymbol{z}}_{t}$ follows the settings of (a)-(d). For the Models
(v)-(viii), we set ${\bf A}=(a_{ij})_{1\leq i,j\leq p}$ with
$a_{ij}\stackrel{{\scriptstyle i.i.d}}{{\sim}}U(-1,1)$ and
${\boldsymbol{z}}_{t}$ follows the settings of (a)-(d).

Here we use parametric bootstrap method to determine the critical value of
$L_{r}$; The empirical sizes of the seven test statistics listed above are
reported in Tables 1 for $K=2$. The simulation results with $K=4,6$ are in the
Supplementary Material. In each table, the results are summarized for Models
(i)-(viii) with different combinations of $n$ and $p$, i.e., $n=100,200$ and
$p=30,60,120,240$. It is easy to see that the degenerate U-statistics $L_{D}$,
$L_{R}$ and $L_{\tau^{*}}$ can control the sizes very well in most of the
cases. However, the empirical sizes of the sum-type statistic $S_{r}$, the
simple linear rank statistic $L_{\rho}$ and non-degenerate U-statistic
$L_{\tau}$ are a little smaller than the nominal level.  The parametric
bootstrap method proposed by Chang et al. (2017) can control the empirical
sizes of $L_{r}$ in most cases. In the supplemental material, we also proposed
a permutation procedure to control the empirical sizes of the above rank-based
test statistics. Additional simulation studies show the good performance of
the permutation procedure.

Table 1:  Sizes of tests with $K=2$ under Model (i)-(viii). $n$ | $p$ | i | ii
---|---|---|---
|  | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$ | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$
100 | 30 | 0.043 | 0.013 | 0.014 | 0.054 | 0.067 | 0.05 | 0.02 | 0.045 | 0.016 | 0.015 | 0.046 | 0.065 | 0.04 | 0.027
100 | 60 | 0.036 | 0.01 | 0.008 | 0.052 | 0.073 | 0.041 | 0.018 | 0.039 | 0.016 | 0.011 | 0.047 | 0.076 | 0.031 | 0.015
100 | 120 | 0.034 | 0.007 | 0.006 | 0.029 | 0.073 | 0.023 | 0.002 | 0.042 | 0.009 | 0.007 | 0.033 | 0.068 | 0.026 | 0.004
100 | 240 | 0.044 | 0.009 | 0.009 | 0.034 | 0.081 | 0.027 | 0 | 0.043 | 0.009 | 0.005 | 0.04 | 0.082 | 0.028 | 0
200 | 30 | 0.035 | 0.017 | 0.016 | 0.045 | 0.05 | 0.04 | 0.034 | 0.056 | 0.016 | 0.015 | 0.04 | 0.056 | 0.035 | 0.042
200 | 60 | 0.038 | 0.022 | 0.023 | 0.045 | 0.054 | 0.046 | 0.025 | 0.042 | 0.014 | 0.016 | 0.034 | 0.044 | 0.03 | 0.027
200 | 120 | 0.039 | 0.012 | 0.008 | 0.032 | 0.056 | 0.025 | 0.015 | 0.046 | 0.015 | 0.018 | 0.04 | 0.044 | 0.034 | 0.008
200 | 240 | 0.043 | 0.008 | 0.01 | 0.037 | 0.06 | 0.032 | 0.005 | 0.047 | 0.018 | 0.014 | 0.051 | 0.067 | 0.046 | 0
|  | iii | iv
100 | 30 | 0.022 | 0.009 | 0.007 | 0.03 | 0.052 | 0.026 | 0.085 | 0.038 | 0.011 | 0.012 | 0.044 | 0.061 | 0.034 | 0.038
100 | 60 | 0.024 | 0.006 | 0.004 | 0.034 | 0.05 | 0.027 | 0.089 | 0.042 | 0.008 | 0.005 | 0.028 | 0.057 | 0.022 | 0.05
100 | 120 | 0.032 | 0.01 | 0.011 | 0.044 | 0.087 | 0.029 | 0.055 | 0.061 | 0.011 | 0.009 | 0.055 | 0.097 | 0.042 | 0.057
100 | 240 | 0.036 | 0.033 | 0.01 | 0.035 | 0.086 | 0.03 | 0.034 | 0.054 | 0.009 | 0.008 | 0.04 | 0.088 | 0.028 | 0.035
200 | 30 | 0.051 | 0.014 | 0.015 | 0.041 | 0.056 | 0.04 | 0.08 | 0.037 | 0.018 | 0.019 | 0.044 | 0.055 | 0.046 | 0.071
200 | 60 | 0.035 | 0.015 | 0.014 | 0.04 | 0.054 | 0.035 | 0.08 | 0.043 | 0.019 | 0.018 | 0.048 | 0.059 | 0.045 | 0.075
200 | 120 | 0.036 | 0.016 | 0.011 | 0.046 | 0.062 | 0.039 | 0.086 | 0.047 | 0.016 | 0.017 | 0.039 | 0.066 | 0.039 | 0.056
200 | 240 | 0.042 | 0.021 | 0.016 | 0.041 | 0.062 | 0.035 | 0.034 | 0.036 | 0.014 | 0.012 | 0.045 | 0.068 | 0.041 | 0.062
|  | v | vi
100 | 30 | 0.044 | 0.018 | 0.017 | 0.049 | 0.067 | 0.041 | 0.031 | 0.051 | 0.017 | 0.012 | 0.049 | 0.067 | 0.043 | 0.033
100 | 60 | 0.038 | 0.015 | 0.013 | 0.05 | 0.082 | 0.044 | 0.015 | 0.043 | 0.011 | 0.012 | 0.042 | 0.066 | 0.039 | 0.016
100 | 120 | 0.035 | 0.008 | 0.006 | 0.034 | 0.073 | 0.026 | 0.001 | 0.036 | 0.011 | 0.007 | 0.035 | 0.076 | 0.025 | 0.004
100 | 240 | 0.044 | 0.012 | 0.009 | 0.045 | 0.097 | 0.036 | 0 | 0.029 | 0.009 | 0.003 | 0.045 | 0.091 | 0.027 | 0
200 | 30 | 0.057 | 0.013 | 0.012 | 0.031 | 0.046 | 0.031 | 0.055 | 0.061 | 0.018 | 0.022 | 0.054 | 0.063 | 0.044 | 0.035
200 | 60 | 0.037 | 0.009 | 0.008 | 0.038 | 0.053 | 0.034 | 0.038 | 0.043 | 0.014 | 0.013 | 0.043 | 0.06 | 0.043 | 0.029
200 | 120 | 0.044 | 0.018 | 0.017 | 0.049 | 0.075 | 0.043 | 0.021 | 0.039 | 0.016 | 0.012 | 0.046 | 0.061 | 0.04 | 0.016
200 | 240 | 0.037 | 0.008 | 0.009 | 0.032 | 0.065 | 0.034 | 0.008 | 0.052 | 0.014 | 0.016 | 0.052 | 0.068 | 0.048 | 0.003
|  | vii | viii
100 | 30 | 0.041 | 0.01 | 0.009 | 0.037 | 0.056 | 0.031 | 0.09 | 0.053 | 0.013 | 0.015 | 0.055 | 0.072 | 0.05 | 0.047
100 | 60 | 0.038 | 0.011 | 0.009 | 0.045 | 0.07 | 0.037 | 0.063 | 0.041 | 0.010 | 0.006 | 0.041 | 0.075 | 0.035 | 0.057
100 | 120 | 0.037 | 0.017 | 0.008 | 0.046 | 0.087 | 0.036 | 0.055 | 0.045 | 0.008 | 0.008 | 0.035 | 0.075 | 0.026 | 0.04
100 | 240 | 0.053 | 0.009 | 0.005 | 0.034 | 0.085 | 0.022 | 0.023 | 0.038 | 0.009 | 0.005 | 0.042 | 0.094 | 0.028 | 0.037
200 | 30 | 0.061 | 0.024 | 0.024 | 0.053 | 0.063 | 0.053 | 0.087 | 0.048 | 0.011 | 0.013 | 0.032 | 0.04 | 0.03 | 0.067
200 | 60 | 0.036 | 0.02 | 0.017 | 0.049 | 0.066 | 0.046 | 0.075 | 0.045 | 0.015 | 0.016 | 0.046 | 0.058 | 0.046 | 0.058
200 | 120 | 0.048 | 0.021 | 0.02 | 0.05 | 0.077 | 0.042 | 0.064 | 0.039 | 0.019 | 0.016 | 0.055 | 0.081 | 0.047 | 0.063
200 | 240 | 0.049 | 0.01 | 0.011 | 0.039 | 0.056 | 0.036 | 0.041 | 0.054 | 0.02 | 0.023 | 0.041 | 0.063 | 0.036 | 0.056

###  3.2 Power comparison

We consider the following eight examples as the data generation procedure in
order to investigate the powers of different test statistics. Let
${\boldsymbol{z}}_{t}\sim N(\boldsymbol{0},{\bf I}_{p})$. In the following,
with slight abuse of notation, we write
$f(v)=\left(f\left(v_{1}\right),\ldots,f\left(v_{p}\right)\right)^{\top}$ for
any univariate function $f:\mathbb{R}\rightarrow\mathbb{R}$ and
$v=\left(v_{1},\ldots,v_{p}\right)^{\top}\in\mathbb{R}^{p}$. That is, (I)
$\boldsymbol{\varepsilon}_{t}={\bf
A}\boldsymbol{\varepsilon}_{t-1}+{\boldsymbol{z}}_{t}$; (II)
$\boldsymbol{\varepsilon}_{t}=\sin(\frac{2\pi}{3}{\bf
A}\boldsymbol{\varepsilon}_{t-1})+{\boldsymbol{z}}_{t}$; (III)
$\boldsymbol{\varepsilon}_{t}=\sin(\frac{\pi}{3}({\bf
A}\boldsymbol{\varepsilon}_{t-1})^{1/3})+{\boldsymbol{z}}_{t}$; (IV)
$\boldsymbol{\varepsilon}_{t}=({\bf
A}\boldsymbol{\varepsilon}_{t-1})^{1/3}+{\boldsymbol{z}}_{t}$; (V)
$\boldsymbol{\varepsilon}_{t}={\boldsymbol{z}}_{t}+{\bf
A}{\boldsymbol{z}}_{t-1}$; (VI)
$\boldsymbol{\varepsilon}_{t}={\boldsymbol{z}}_{t}+\sin(\frac{2\pi}{3}{\bf
A}{\boldsymbol{z}}_{t-1})$; (VII)
$\boldsymbol{\varepsilon}_{t}={\boldsymbol{z}}_{t}+\sin(\frac{\pi}{3}({\bf
A}{\boldsymbol{z}}_{t-1})^{1/3})$; (VIII)
$\boldsymbol{\varepsilon}_{t}={\boldsymbol{z}}_{t}+({\bf
A}{\boldsymbol{z}}_{t-1})^{1/3}$.

We consider ${\bf A}=(a_{ij})_{1\leq i,j\leq p}$ with $a_{ij}\sim
U(-\rho,\rho)$ if $1\leq i,j\leq k_{0}$ and $a_{ij}=0$ otherwise. Models (I)
and (V) can be classified as the linear relationship in autocorrelations,
while Models (IV) and (VIII) can be classified as the monotone relationship in
autocorrelations. The rest of the models are the non-linear and non-monotone
relationships in autocorrelation.

Figures 1 report the power curves with different $\rho$ for $K=2$. The power
curves with $K=4,6$ are in the Supplementary Material. For these three
figures, we set $k_{0}=2$, $n=100$ and $p=30$. It is clear that the parameter
$\rho$ controls the level of the autocorrelation. Therefore, as $\rho$
increasing, the power curves show the upward trend as well for most of the
models. Moreover, the power curves of the degenerate U-statistics $L_{D}$,
$L_{R}$ and $L_{\tau^{*}}$ are higher than that of other max-type test
statistics, i.e., $L_{r}$, $L_{\rho}$ and $L_{\tau}$. The sum-type test
statistic $S_{r}$ has the lowest power curve in most cases. It is not
surprising because $k_{0}$ was set as 2 here and sum-type test cannot work
well under the sparse alternatives.

Figure 2 shows the power curves with different $k_{0}$. For fixed $p$ and
$\rho$, the parameter $k_{0}$ is used to control the sparsity of the
autocorrelations. The higher value of $k_{0}$ yields the lower level of the
sparsity in the autocorrelations. As expected, the power curves of the max-
type test statistics, i.e., $L_{r}$, $L_{\rho}$, $L_{\tau}$, $L_{D}$, $L_{R}$
and $L_{\tau^{*}}$ have the downward trend when $k_{0}$ increasing in most of
the models. Moreover, among the six max-type test statistics, the power curves
of the degenerate U-statistics $L_{D}$, $L_{R}$ and $L_{\tau^{*}}$ are
relatively higher than that of the other three test statistics. In contrast,
the power curve of the sum-type test statistic $S_{r}$ has the upward trend as
$k_{0}$ increasing in most models.

Figure 3 shows the power curves with different $p$. For fixed $\rho$ and
$k_{0}$, as the parameter $p$ increasing, the signal strength tends to
decrease. Therefore, it is not surprising that all power curves of the seven
test statistics show the downward trend as $p$ increasing. The power curves of
degenerate U-statistics $L_{D}$, $L_{R}$ and $L_{\tau^{*}}$ are the highest in
the seven test statistics. In contrast, the power curve of the sum-type test
statistic $S_{r}$ is the lowest among the seven statistics.

Figure 1: Power curves of different methods with different $\rho$ and
$k_{0}=2,n=100,p=30,K=2$. Figure 2: Power curves of different methods with
different $k_{0}$ and $\rho=0.6,n=100,p=30,K=2$. Figure 3: Power curves of
different methods with different $p$ and $\rho=0.6,n=100,k_{0}=2,K=2$.

##  4 Conclusion

To test the high-dimensional white noise, we develop the max-type tests based
on three families of rank based statistics, including the simple linear rank
statistics, non-degenerate U-statistics and degenerate U-statistics. The
proposed tests are distribution free and in particular, the degenerate
U-statistics can be used to detect the non-linear and non-monotone
relationships in autocorrelations. Finally, as the theoretical contribution of
this paper, we have relaxed the cross-sectional independence assumption in
existing literature when deriving the asymptotic distributions for the rank
correlation statistics.  From the simulation studies, we found that the power
of degenerate U-statistics $L_{D},L_{R},L_{\tau^{*}}$ have the best
performance. So we suggest the degenerate U-statistics proposed in subsection
2.3 in practice.

For the future directions related to the high-dimensional white noise test, it
is also important to develop the theory for the sum-type tests based on the
rank based statistics. The asymptotic independence between the max-type test
and sum-type test based on the rank based statistics is also necessary to be
established because of its usefulness in constructing some combination test
which can be robust to both sparse and dense alternatives.

## Acknowledgement

The research of Dachuan Chen is supported by the National Natural Science
Foundation of China (Grants 12101335 and 12271271), the Natural Science
Foundation of Tianjin (Grant 21JCQNJC00020), the Fundamental Research Funds
for the Central Universities, Nankai University (Grants 63211088, 63221050,
and 63231013) and Wukong Investment Research Funds. Long Feng was partially
supported by Shenzhen Wukong Investment Company, the Fundamental Research
Funds for the Central Universities under Grant No. ZB22000105 and 63233075,
the China National Key R&D Program (Grant Nos. 2019YFC1908502, 2022YFA1003703,
2022YFA1003802, 2022YFA1003803) and the National Natural Science Foundation of
China Grants (Nos. 12271271, 11925106, 12231011, 11931001 and 11971247).
Fengyi Song and Long Feng are co-corresponding authors and equally contributed
to this paper.

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School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai
University E-mail<EMAIL_ADDRESS>
School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai
University E-mail<EMAIL_ADDRESS>
School of Statistics and Data Science, KLMDASR, LEBPS, and LPMC, Nankai
University E-mail<EMAIL_ADDRESS>
Supplement to

“Rank Based Tests for High Dimensional White Noise”

Dachuan Chen, Fengyi Song and Long Feng

Nankai University

##  5 Chatterjee’s rank Correlation

For a pair of continuous random variables $(X,Y)$, Chatterjee (2021) proposed
a new rank correlation, i.e.

$\displaystyle\xi_{n}(X,Y)=1-\frac{3\sum_{i=1}^{n-1}|r_{i+1}-r_{i}|}{n^{2}-1}$

where $r_{i}$ is the rank of $Y_{(i)}$. Here we rearrange the data as
$(X_{(1)},Y_{(1)}),\cdots,(X_{(n)},Y_{(n)})$ such that $X_{(1)}\leq\cdots\leq
X_{(n)}$. So, we can also consider the test based on Chatterjee’s rank
correlation of the form

$\displaystyle\Xi_{ij}(k)=1-\frac{3\sum_{t=1}^{n-k}|R_{n-k,t+k+1}^{ij}(k)-R_{n-k,t+k}^{ij}(k)|}{(n-k+1)^{2}-1}$
(5.15)

By Theorem 2.2 in Chatterjee (2021),
$\sqrt{n-k+1}\Xi_{ij}(k)\mathop{\rightarrow}\limits^{d}N(0,2/5)$ as
$n\to\infty$ under the null hypothesis. Thus, we propose the following
statistics for testing $H_{0}$:

$\displaystyle\Xi_{n}=\max_{1\leq k\leq K}\max_{1\leq i,j\leq
p}\sqrt{\frac{5(n-k+1)}{2}}|\Xi_{ij}(k)|$ (5.16)

Next, we state the theoretical result about the limiting null distribution of
$\Xi_{n}$.

######  Theorem 10.

If $(\varepsilon_{t1},\cdots,\varepsilon_{tp})$ are mutually independent
continuous random variables, under $H_{0}$, for any $y\in\mathbb{R}$, we have

$\displaystyle P\left(\Xi_{n}^{2}-2\log(N)+\log\log(N)\leq
y\right)=\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}+o(1)$

as $n,p\to\infty$.

Based on Theorem S1, we proposed the following high dimensional white noise
test based on Chatterjee’s Correlation

$T_{\alpha}^{\xi}\doteq I\left(\Xi_{n}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\geq
q_{\alpha}\right),$ (5.17)

where $q_{\alpha}=-\log(\pi)-2\log\log(1-\alpha)^{-1}$.

Table 2 show the empirical sizes of the proposed test $T_{\alpha}^{\xi}$ under
the same settings as subsection 3.1 in the main text. We observe that the
empirical sizes of $T_{\alpha}^{\xi}$ is a little conservative in most cases.
Additionally, we also show the power of the proposed test $T_{\alpha}^{\xi}$
and the seven tests in the main text in Table 3 with
$n=100,p=30,\rho=0.5,k_{0}=2$. The other settings are all the same as
subsection 3.2 in the main text. We found that $T_{\alpha}^{\xi}$ does not
perform very well in most cases, which is consistent with many recent studies
(Cao and Bickel, 2020; Shi et al., 2021; Lin and Han, 2023). They showed that
independence tests based on Chatterjee’s rank correlation are unfortunately
rate-inefficient against various local alternatives.

Table 2:  Empirical sizes of $L_{\xi}$ under Models (i)-(viii). ($L_{\xi}$: the max-type test defined in (5.17).) | Models | i | ii | iii | iv | v | vi | vii | viii
---|---|---|---|---|---|---|---|---|---
$n$ | $p$ | $K=2$
100 | 30 | 0.022 | 0.03 | 0.026 | 0.029 | 0.026 | 0.032 | 0.022 | 0.019
100 | 60 | 0.026 | 0.019 | 0.028 | 0.026 | 0.027 | 0.02 | 0.019 | 0.024
100 | 120 | 0.032 | 0.019 | 0.023 | 0.023 | 0.027 | 0.025 | 0.031 | 0.022
100 | 240 | 0.031 | 0.018 | 0.02 | 0.027 | 0.03 | 0.022 | 0.023 | 0.019
200 | 30 | 0.033 | 0.025 | 0.039 | 0.03 | 0.039 | 0.03 | 0.038 | 0.035
200 | 60 | 0.033 | 0.029 | 0.031 | 0.03 | 0.029 | 0.032 | 0.035 | 0.028
200 | 120 | 0.031 | 0.030 | 0.029 | 0.029 | 0.035 | 0.037 | 0.029 | 0.038
200 | 240 | 0.030 | 0.035 | 0.039 | 0.027 | 0.025 | 0.033 | 0.038 | 0.027
|  | $K=4$
100 | 30 | 0.029 | 0.022 | 0.023 | 0.022 | 0.025 | 0.031 | 0.032 | 0.016
100 | 60 | 0.016 | 0.022 | 0.021 | 0.025 | 0.02 | 0.031 | 0.028 | 0.022
100 | 120 | 0.018 | 0.023 | 0.032 | 0.015 | 0.029 | 0.019 | 0.029 | 0.028
100 | 240 | 0.03 | 0.023 | 0.03 | 0.029 | 0.018 | 0.02 | 0.031 | 0.031
200 | 30 | 0.036 | 0.033 | 0.038 | 0.033 | 0.04 | 0.032 | 0.032 | 0.033
200 | 60 | 0.04 | 0.03 | 0.023 | 0.032 | 0.023 | 0.037 | 0.028 | 0.023
200 | 120 | 0.022 | 0.033 | 0.023 | 0.032 | 0.02 | 0.028 | 0.034 | 0.023
200 | 240 | 0.02 | 0.038 | 0.035 | 0.036 | 0.028 | 0.024 | 0.022 | 0.02
|  | $K=6$
100 | 30 | 0.03 | 0.027 | 0.022 | 0.023 | 0.027 | 0.018 | 0.016 | 0.028
100 | 60 | 0.015 | 0.028 | 0.028 | 0.032 | 0.03 | 0.03 | 0.032 | 0.018
100 | 120 | 0.021 | 0.022 | 0.021 | 0.022 | 0.029 | 0.021 | 0.018 | 0.021
100 | 240 | 0.021 | 0.027 | 0.025 | 0.024 | 0.016 | 0.016 | 0.022 | 0.029
200 | 30 | 0.023 | 0.028 | 0.02 | 0.013 | 0.026 | 0.027 | 0.024 | 0.032
200 | 60 | 0.012 | 0.022 | 0.022 | 0.012 | 0.02 | 0.019 | 0.017 | 0.018
200 | 120 | 0.021 | 0.018 | 0.02 | 0.012 | 0.02 | 0.011 | 0.019 | 0.018
200 | 240 | 0.023 | 0.021 | 0.012 | 0.010 | 0.022 | 0.012 | 0.025 | 0.022
Table 3:  Power of tests with $n=100,p=30,\rho=0.5,k_{0}=2$ under Models (I)-(VIII). ($L_{\xi}$: the max-type test defined in (5.17).) | Methods
---|---
Models | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$ | $L_{\xi}$
| $K=2$
I | 0.22 | 0.22 | 0.22 | 0.38 | 0.42 | 0.37 | 0.22 | 0.03
II | 0.41 | 0.43 | 0.44 | 0.81 | 0.83 | 0.79 | 0.43 | 0.02
III | 0.45 | 0.48 | 0.48 | 0.79 | 0.8 | 0.8 | 0.41 | 0.06
IV | 0.64 | 0.63 | 0.64 | 0.88 | 0.9 | 0.88 | 0.49 | 0.21
V | 0.15 | 0.1 | 0.12 | 0.26 | 0.27 | 0.25 | 0.05 | 0.02
VI | 0.29 | 0.26 | 0.23 | 0.45 | 0.46 | 0.44 | 0.1 | 0.01
VII | 0.38 | 0.33 | 0.37 | 0.54 | 0.55 | 0.55 | 0.16 | 0.06
VIII | 0.26 | 0.24 | 0.23 | 0.42 | 0.49 | 0.43 | 0.17 | 0.02
| $K=4$
I | 0.21 | 0.21 | 0.2 | 0.37 | 0.42 | 0.34 | 0.03 | 0.02
II | 0.42 | 0.46 | 0.44 | 0.7 | 0.75 | 0.71 | 0.12 | 0.08
III | 0.44 | 0.42 | 0.42 | 0.76 | 0.81 | 0.75 | 0.16 | 0.03
IV | 0.58 | 0.56 | 0.56 | 0.83 | 0.86 | 0.83 | 0.28 | 0.1
V | 0.11 | 0.06 | 0.06 | 0.21 | 0.26 | 0.19 | 0 | 0.04
VI | 0.17 | 0.17 | 0.17 | 0.37 | 0.38 | 0.36 | 0.03 | 0
VII | 0.18 | 0.18 | 0.18 | 0.47 | 0.51 | 0.45 | 0.06 | 0.01
VIII | 0.13 | 0.16 | 0.16 | 0.39 | 0.42 | 0.38 | 0.04 | 0.03
| $K=6$
I | 0.19 | 0.17 | 0.17 | 0.36 | 0.42 | 0.35 | 0.04 | 0.05
II | 0.31 | 0.39 | 0.42 | 0.71 | 0.72 | 0.68 | 0.08 | 0.07
III | 0.42 | 0.45 | 0.45 | 0.75 | 0.78 | 0.75 | 0.02 | 0.01
IV | 0.49 | 0.51 | 0.53 | 0.87 | 0.88 | 0.87 | 0.12 | 0.12
V | 0.1 | 0.07 | 0.06 | 0.13 | 0.19 | 0.12 | 0 | 0.04
VI | 0.07 | 0.11 | 0.09 | 0.26 | 0.3 | 0.24 | 0 | 0.03
VII | 0.2 | 0.19 | 0.2 | 0.38 | 0.38 | 0.37 | 0.02 | 0.04
VIII | 0.14 | 0.18 | 0.16 | 0.31 | 0.34 | 0.31 | 0.01 | 0.02

##  6 L-statistic

As shown in the main text, the max-type test statistics performs very well
under sparse alternative. Motivated by Chang et al. (2023), we consider an
L-statistic for high dimensional white noise test, which combines the first
several largest signals together. That is,

$\displaystyle L_{V}=\sum_{l=1}^{L}V_{(l)},~{}~{}L_{U}=\sum_{l=1}^{L}U_{(l)}$
(6.18)

where $V_{(l)}$ and $U_{(l)}$ are the $l$-th largest maximum of
$\\{|V_{ij}(k)|\\}_{1\leq i,j\leq p,1\leq k\leq K}$ and
$\\{|U_{ij}(k)|\\}_{1\leq i,j\leq p,1\leq k\leq K}$, respectively.

It is difficult to establish the limit null distribution of $L_{V}$ and
$L_{U}$. So we adopt the permutation test to calculate the critical value of
each test. We randomly rearrange
$\\{\varepsilon_{1},\cdots,\varepsilon_{T}\\}$ as
$\\{\varepsilon_{\pi(1)},\cdots,\varepsilon_{\pi(T)}\\}$ where $\pi$ is a
permutation of $\\{1,\cdots,T\\}$. The permutation test statistic
$\tilde{L}_{V}$ and $\tilde{L}_{U}$ are accordingly built from the permutation
sample $\\{\varepsilon_{\pi(1)},\cdots,\varepsilon_{\pi(T)}\\}$. When this
procedure is repeated many times, the permutation critical value
$z_{\alpha}^{L}$ and $z_{\alpha}^{U}$ are the empirical $1-\alpha$ quantile of
the permutation test statistic, respectively. The tests with rejection region
$L_{V}\geq z_{\alpha}^{L}$ and $L_{U}\geq z_{\alpha}^{U}$ are our proposal.

Here we give a simulation study of the $L$-statistics. Let $\tilde{L}_{\tau}$,
$\tilde{L}_{\rho}$, $\tilde{L}_{\tau^{*}}$, $\tilde{L}_{D}$, $\tilde{L}_{R}$
denote the corresponding $L$-statistics based on Kendall’s tau, Spearman’s
rho, Bergsma-Dassios-Yanagimoto’s $\tau^{*}$, Hoeffding’s $D$, Blum-Kiefer-
Rosenblatt’s $R$, respectively. Table 4 reports the empirical sizes of the
above test statistics with $K=2$ and different $L$ under Model (i). We found
that the permutation procedure can control all the empirical sizes of these
tests very well. To show the performance of $L$-statistics, we consider a
power comparison of $\tilde{L}_{\tau^{*}}$ with different $L$ because the
Bergsma-Dassios-Yanagimoto’s $\tau^{*}$ statistic performs very well in most
cases in the simulation studies in the main text. We consider the same
settings as subsection 3.2 in the main text except that $k_{0}=1,\cdots,10$
and $\rho=0.98k_{0}^{2/3}$. From Figure 4, we observe that when the number of
non-zero autocorrelations is small, $\tilde{L}_{\tau^{*}}$ with small $L$s
have better performance than large $L$s and vice versa. So the optimal $L$
depends on the sparsity of the autocorrelations. Generally speaking,
$\tilde{L}_{\tau^{*}}$ with $L>1$ outperforms $\tilde{L}_{\tau^{*}}$ with
$L=1$, i.e. $L_{\tau^{*}}$ in most cases. So how to derive the limit null
distribution of rank based $L$-statistics and choose the optimal $L$ for high
dimensional white noise test deserves further studies.

Table 4:  Sizes of $L$-statistic tests with $K=2$ under Model (i). |  $L$
---|---
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
|  $(n,p)=(100,30)$
$\tilde{L}_{\tau}$ | 0.041 | 0.046 | 0.059 | 0.052 | 0.046 | 0.058 | 0.05 | 0.052 | 0.059 | 0.048
$\tilde{L}_{\rho}$ | 0.046 | 0.059 | 0.056 | 0.048 | 0.052 | 0.059 | 0.042 | 0.05 | 0.05 | 0.054
$\tilde{L}_{\tau^{*}}$ | 0.059 | 0.046 | 0.05 | 0.043 | 0.047 | 0.045 | 0.047 | 0.047 | 0.056 | 0.05
$\tilde{L}_{D}$ | 0.046 | 0.045 | 0.05 | 0.05 | 0.059 | 0.043 | 0.05 | 0.058 | 0.045 | 0.043
$\tilde{L}_{R}$ | 0.054 | 0.051 | 0.059 | 0.052 | 0.056 | 0.048 | 0.045 | 0.056 | 0.054 | 0.045
|  $(n,p)=(100,60)$
$\tilde{L}_{\tau}$ | 0.054 | 0.051 | 0.043 | 0.05 | 0.04 | 0.054 | 0.054 | 0.04 | 0.046 | 0.057
$\tilde{L}_{\rho}$ | 0.044 | 0.04 | 0.057 | 0.047 | 0.048 | 0.051 | 0.04 | 0.054 | 0.055 | 0.048
$\tilde{L}_{\tau^{*}}$ | 0.04 | 0.06 | 0.048 | 0.041 | 0.052 | 0.049 | 0.056 | 0.045 | 0.056 | 0.058
$\tilde{L}_{D}$ | 0.04 | 0.059 | 0.056 | 0.049 | 0.044 | 0.049 | 0.058 | 0.042 | 0.041 | 0.058
$\tilde{L}_{R}$ | 0.046 | 0.044 | 0.056 | 0.058 | 0.047 | 0.04 | 0.048 | 0.05 | 0.043 | 0.05
|  $(n,p)=(100,90)$
$\tilde{L}_{\tau}$ | 0.047 | 0.041 | 0.053 | 0.048 | 0.058 | 0.049 | 0.048 | 0.043 | 0.041 | 0.049
$\tilde{L}_{\rho}$ | 0.047 | 0.057 | 0.042 | 0.057 | 0.046 | 0.058 | 0.055 | 0.043 | 0.059 | 0.052
$\tilde{L}_{\tau^{*}}$ | 0.058 | 0.053 | 0.041 | 0.048 | 0.053 | 0.049 | 0.045 | 0.053 | 0.051 | 0.041
$\tilde{L}_{D}$ | 0.054 | 0.054 | 0.052 | 0.05 | 0.044 | 0.057 | 0.047 | 0.05 | 0.06 | 0.06
$\tilde{L}_{R}$ | 0.051 | 0.057 | 0.06 | 0.056 | 0.057 | 0.054 | 0.049 | 0.057 | 0.058 | 0.049
|  $(n,p)=(100,120)$
$\tilde{L}_{\tau}$ | 0.052 | 0.049 | 0.049 | 0.057 | 0.049 | 0.043 | 0.041 | 0.048 | 0.045 | 0.04
$\tilde{L}_{\rho}$ | 0.058 | 0.054 | 0.044 | 0.049 | 0.056 | 0.058 | 0.06 | 0.049 | 0.049 | 0.045
$\tilde{L}_{\tau^{*}}$ | 0.044 | 0.05 | 0.059 | 0.059 | 0.049 | 0.06 | 0.055 | 0.05 | 0.052 | 0.049
$\tilde{L}_{D}$ | 0.058 | 0.059 | 0.044 | 0.044 | 0.054 | 0.041 | 0.05 | 0.045 | 0.041 | 0.044
$\tilde{L}_{R}$ | 0.06 | 0.05 | 0.053 | 0.055 | 0.046 | 0.049 | 0.042 | 0.045 | 0.047 | 0.042
Figure 4: Power curves of different $L$-statistics with different $L$ and
$n=100,p=30,K=2$ under Model I.

##  7 Additional Simulation Results of Section 3 in the main document

The empirical sizes of the seven test statistics listed in the beginning of
Section 3 in the main document are reported in Table 5 and 6, respectively.
Figures 5 and 6 report the power curves of the seven test statistics with
different $\rho$ for $K=4$ and $6$, respectively.

Table 5:  Sizes of tests with $K=4$ under Model (i)-(viii). $n$ | $p$ | i | ii
---|---|---|---
|  | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$ | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$
100 | 30 | 0.042 | 0.016 | 0.016 | 0.055 | 0.087 | 0.047 | 0.013 | 0.038 | 0.007 | 0.007 | 0.04 | 0.074 | 0.037 | 0.014
100 | 60 | 0.037 | 0.012 | 0.008 | 0.04 | 0.07 | 0.036 | 0.001 | 0.031 | 0.006 | 0.005 | 0.031 | 0.073 | 0.03 | 0.002
100 | 120 | 0.036 | 0.011 | 0.008 | 0.036 | 0.09 | 0.026 | 0 | 0.027 | 0.009 | 0.004 | 0.04 | 0.075 | 0.03 | 0
100 | 240 | 0.029 | 0.009 | 0.001 | 0.037 | 0.091 | 0.018 | 0 | 0.024 | 0.006 | 0.004 | 0.028 | 0.08 | 0.021 | 0
200 | 30 | 0.047 | 0.017 | 0.014 | 0.047 | 0.055 | 0.041 | 0.022 | 0.051 | 0.016 | 0.015 | 0.036 | 0.047 | 0.034 | 0.034
200 | 60 | 0.031 | 0.016 | 0.013 | 0.036 | 0.048 | 0.028 | 0.012 | 0.037 | 0.014 | 0.01 | 0.038 | 0.058 | 0.035 | 0.012
200 | 120 | 0.029 | 0.013 | 0.013 | 0.042 | 0.057 | 0.038 | 0.002 | 0.033 | 0.015 | 0.016 | 0.047 | 0.066 | 0.04 | 0.001
200 | 240 | 0.031 | 0.018 | 0.014 | 0.038 | 0.059 | 0.036 | 0 | 0.028 | 0.008 | 0.005 | 0.045 | 0.066 | 0.033 | 0
|  | iii | iv
100 | 30 | 0.047 | 0.011 | 0.011 | 0.039 | 0.06 | 0.035 | 0.06 | 0.052 | 0.008 | 0.006 | 0.04 | 0.071 | 0.036 | 0.036
100 | 60 | 0.036 | 0.013 | 0.009 | 0.039 | 0.077 | 0.034 | 0.044 | 0.045 | 0.009 | 0.009 | 0.056 | 0.092 | 0.041 | 0.033
100 | 120 | 0.038 | 0.01 | 0.008 | 0.031 | 0.081 | 0.025 | 0.028 | 0.047 | 0.011 | 0.009 | 0.039 | 0.094 | 0.03 | 0.023
100 | 240 | 0.028 | 0.004 | 0.003 | 0.027 | 0.082 | 0.02 | 0.008 | 0.051 | 0.006 | 0.003 | 0.037 | 0.104 | 0.021 | 0.015
200 | 30 | 0.031 | 0.017 | 0.018 | 0.04 | 0.049 | 0.036 | 0.06 | 0.057 | 0.017 | 0.016 | 0.045 | 0.057 | 0.037 | 0.059
200 | 60 | 0.042 | 0.014 | 0.012 | 0.038 | 0.047 | 0.034 | 0.056 | 0.053 | 0.015 | 0.012 | 0.043 | 0.057 | 0.032 | 0.045
200 | 120 | 0.034 | 0.014 | 0.008 | 0.037 | 0.057 | 0.036 | 0.046 | 0.038 | 0.014 | 0.018 | 0.041 | 0.056 | 0.037 | 0.035
200 | 240 | 0.030 | 0.01 | 0.01 | 0.043 | 0.074 | 0.033 | 0.012 | 0.048 | 0.012 | 0.012 | 0.045 | 0.072 | 0.041 | 0.029
|  | v | vi
100 | 30 | 0.039 | 0.011 | 0.01 | 0.038 | 0.071 | 0.041 | 0.013 | 0.039 | 0.013 | 0.011 | 0.039 | 0.072 | 0.029 | 0.026
100 | 60 | 0.041 | 0.013 | 0.014 | 0.055 | 0.09 | 0.047 | 0.003 | 0.033 | 0.007 | 0.005 | 0.034 | 0.067 | 0.03 | 0.001
100 | 120 | 0.031 | 0.016 | 0.006 | 0.044 | 0.086 | 0.036 | 0.001 | 0.032 | 0.013 | 0.007 | 0.037 | 0.083 | 0.027 | 0
100 | 240 | 0.027 | 0.006 | 0.004 | 0.025 | 0.08 | 0.019 | 0 | 0.034 | 0.008 | 0.005 | 0.035 | 0.106 | 0.029 | 0
200 | 30 | 0.053 | 0.024 | 0.023 | 0.049 | 0.064 | 0.043 | 0.027 | 0.052 | 0.017 | 0.016 | 0.047 | 0.051 | 0.042 | 0.031
200 | 60 | 0.044 | 0.013 | 0.015 | 0.029 | 0.053 | 0.03 | 0.016 | 0.039 | 0.014 | 0.014 | 0.043 | 0.06 | 0.038 | 0.017
200 | 120 | 0.032 | 0.023 | 0.021 | 0.053 | 0.077 | 0.049 | 0.011 | 0.031 | 0.015 | 0.013 | 0.047 | 0.066 | 0.038 | 0.003
200 | 240 | 0.029 | 0.008 | 0.006 | 0.029 | 0.067 | 0.024 | 0 | 0.029 | 0.011 | 0.009 | 0.032 | 0.053 | 0.029 | 0
|  | vii | viii
100 | 30 | 0.053 | 0.008 | 0.007 | 0.038 | 0.059 | 0.034 | 0.052 | 0.053 | 0.01 | 0.013 | 0.043 | 0.07 | 0.04 | 0.037
100 | 60 | 0.042 | 0.01 | 0.007 | 0.042 | 0.071 | 0.038 | 0.041 | 0.044 | 0.012 | 0.01 | 0.039 | 0.067 | 0.026 | 0.031
100 | 120 | 0.037 | 0.012 | 0.006 | 0.035 | 0.083 | 0.02 | 0.023 | 0.031 | 0.009 | 0.004 | 0.028 | 0.073 | 0.019 | 0.016
100 | 240 | 0.026 | 0.009 | 0.003 | 0.036 | 0.101 | 0.025 | 0.006 | 0.043 | 0.005 | 0.005 | 0.028 | 0.108 | 0.017 | 0.017
200 | 30 | 0.043 | 0.025 | 0.025 | 0.052 | 0.062 | 0.049 | 0.073 | 0.046 | 0.016 | 0.014 | 0.045 | 0.057 | 0.044 | 0.047
200 | 60 | 0.036 | 0.016 | 0.015 | 0.046 | 0.057 | 0.041 | 0.046 | 0.033 | 0.011 | 0.01 | 0.054 | 0.058 | 0.047 | 0.024
200 | 120 | 0.031 | 0.01 | 0.008 | 0.034 | 0.05 | 0.032 | 0.022 | 0.043 | 0.015 | 0.012 | 0.043 | 0.066 | 0.033 | 0.036
200 | 240 | 0.028 | 0.012 | 0.01 | 0.04 | 0.059 | 0.042 | 0.005 | 0.034 | 0.015 | 0.024 | 0.039 | 0.053 | 0.026 | 0.034
Table 6:  Sizes of tests with $K=6$ under Model (i)-(viii). $n$ | $p$ | i | ii
---|---|---|---
|  | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$ | $L_{r}$ | $L_{\tau}$ | $L_{\rho}$ | $L_{\tau^{*}}$ | $L_{D}$ | $L_{R}$ | $S_{r}$
100 | 30 | 0.049 | 0.016 | 0.012 | 0.051 | 0.082 | 0.046 | 0.002 | 0.055 | 0.013 | 0.014 | 0.041 | 0.066 | 0.038 | 0.004
100 | 60 | 0.047 | 0.009 | 0.006 | 0.038 | 0.078 | 0.032 | 0.001 | 0.053 | 0.012 | 0.01 | 0.046 | 0.099 | 0.035 | 0
100 | 120 | 0.036 | 0.016 | 0.011 | 0.053 | 0.105 | 0.038 | 0 | 0.043 | 0.008 | 0.003 | 0.028 | 0.083 | 0.027 | 0
100 | 240 | 0.034 | 0.005 | 0.002 | 0.023 | 0.077 | 0.014 | 0 | 0.041 | 0.007 | 0.002 | 0.026 | 0.095 | 0.018 | 0
200 | 30 | 0.055 | 0.011 | 0.013 | 0.043 | 0.047 | 0.04 | 0.015 | 0.045 | 0.015 | 0.015 | 0.036 | 0.046 | 0.032 | 0.013
200 | 60 | 0.056 | 0.016 | 0.014 | 0.037 | 0.051 | 0.036 | 0.002 | 0.042 | 0.012 | 0.01 | 0.046 | 0.054 | 0.046 | 0.008
200 | 120 | 0.037 | 0.009 | 0.007 | 0.045 | 0.062 | 0.036 | 0 | 0.041 | 0.014 | 0.012 | 0.032 | 0.053 | 0.032 | 0
200 | 240 | 0.039 | 0.019 | 0.013 | 0.043 | 0.071 | 0.04 | 0 | 0.036 | 0.01 | 0.007 | 0.046 | 0.078 | 0.041 | 0
|  | iii | iv
100 | 30 | 0.058 | 0.01 | 0.01 | 0.041 | 0.071 | 0.034 | 0.043 | 0.046 | 0.011 | 0.007 | 0.037 | 0.074 | 0.024 | 0.021
100 | 60 | 0.046 | 0.008 | 0.007 | 0.036 | 0.071 | 0.027 | 0.02 | 0.048 | 0.01 | 0.007 | 0.053 | 0.091 | 0.038 | 0.015
100 | 120 | 0.048 | 0.009 | 0.005 | 0.038 | 0.089 | 0.029 | 0.009 | 0.039 | 0.011 | 0.008 | 0.03 | 0.081 | 0.023 | 0.009
100 | 240 | 0.039 | 0.007 | 0.006 | 0.023 | 0.087 | 0.018 | 0.003 | 0.041 | 0.01 | 0.007 | 0.034 | 0.096 | 0.021 | 0.001
200 | 30 | 0.042 | 0.009 | 0.009 | 0.034 | 0.05 | 0.035 | 0.053 | 0.038 | 0.017 | 0.019 | 0.044 | 0.058 | 0.039 | 0.04
200 | 60 | 0.048 | 0.015 | 0.012 | 0.039 | 0.055 | 0.031 | 0.043 | 0.065 | 0.014 | 0.012 | 0.038 | 0.055 | 0.035 | 0.036
200 | 120 | 0.051 | 0.01 | 0.006 | 0.037 | 0.058 | 0.032 | 0.012 | 0.057 | 0.008 | 0.007 | 0.033 | 0.056 | 0.03 | 0.028
200 | 240 | 0.054 | 0.021 | 0.018 | 0.045 | 0.076 | 0.042 | 0.003 | 0.049 | 0.013 | 0.01 | 0.031 | 0.06 | 0.03 | 0.014
|  | v | vi
100 | 30 | 0.049 | 0.014 | 0.01 | 0.036 | 0.06 | 0.032 | 0.007 | 0.049 | 0.018 | 0.01 | 0.05 | 0.082 | 0.046 | 0.007
100 | 60 | 0.038 | 0.01 | 0.007 | 0.043 | 0.072 | 0.033 | 0.002 | 0.036 | 0.01 | 0.01 | 0.037 | 0.076 | 0.029 | 0
100 | 120 | 0.032 | 0.012 | 0.007 | 0.033 | 0.086 | 0.026 | 0 | 0.032 | 0.009 | 0.008 | 0.026 | 0.069 | 0.022 | 0
100 | 240 | 0.031 | 0.011 | 0.006 | 0.028 | 0.093 | 0.021 | 0 | 0.033 | 0.008 | 0.003 | 0.035 | 0.096 | 0.019 | 0
200 | 30 | 0.042 | 0.019 | 0.014 | 0.044 | 0.065 | 0.04 | 0.02 | 0.047 | 0.017 | 0.014 | 0.038 | 0.047 | 0.036 | 0.021
200 | 60 | 0.041 | 0.016 | 0.014 | 0.051 | 0.071 | 0.043 | 0.006 | 0.035 | 0.014 | 0.011 | 0.044 | 0.06 | 0.037 | 0.001
200 | 120 | 0.032 | 0.014 | 0.011 | 0.034 | 0.053 | 0.03 | 0 | 0.041 | 0.006 | 0.005 | 0.033 | 0.049 | 0.031 | 0
200 | 240 | 0.037 | 0.012 | 0.007 | 0.035 | 0.065 | 0.029 | 0 | 0.037 | 0.013 | 0.013 | 0.04 | 0.073 | 0.032 | 0
|  | vii | viii
100 | 30 | 0.052 | 0.016 | 0.014 | 0.045 | 0.079 | 0.04 | 0.025 | 0.051 | 0.01 | 0.011 | 0.042 | 0.081 | 0.035 | 0.025
100 | 60 | 0.038 | 0.007 | 0.008 | 0.041 | 0.084 | 0.035 | 0.013 | 0.042 | 0.01 | 0.008 | 0.03 | 0.069 | 0.026 | 0.015
100 | 120 | 0.036 | 0.008 | 0.005 | 0.03 | 0.078 | 0.02 | 0.005 | 0.038 | 0.005 | 0.001 | 0.031 | 0.084 | 0.025 | 0.005
100 | 240 | 0.033 | 0.008 | 0.006 | 0.032 | 0.103 | 0.022 | 0 | 0.033 | 0.003 | 0.002 | 0.028 | 0.101 | 0.018 | 0.005
200 | 30 | 0.046 | 0.022 | 0.021 | 0.044 | 0.059 | 0.044 | 0.052 | 0.047 | 0.016 | 0.015 | 0.048 | 0.058 | 0.04 | 0.027
200 | 60 | 0.043 | 0.017 | 0.013 | 0.041 | 0.063 | 0.038 | 0.033 | 0.052 | 0.016 | 0.015 | 0.054 | 0.068 | 0.046 | 0.041
200 | 120 | 0.038 | 0.015 | 0.014 | 0.032 | 0.061 | 0.027 | 0.016 | 0.033 | 0.014 | 0.012 | 0.042 | 0.066 | 0.034 | 0.016
200 | 240 | 0.037 | 0.013 | 0.008 | 0.047 | 0.07 | 0.041 | 0.001 | 0.037 | 0.01 | 0.011 | 0.039 | 0.066 | 0.031 | 0.013
Figure 5: Power curves of different methods with different $\rho$ and
$k_{0}=2,n=100,p=30,K=4$. Figure 6: Power curves of different methods with
different $\rho$ and $k_{0}=2,n=100,p=30,K=6$.

##  8 Proof of Theorems

###  8.1 Proof of Theorems of Simple Linear Rank Statistics

######  Lemma 1.

Suppose that $X,Y$ are two independent continuous random variables. Let
$X_{1},\ldots,X_{n}$ and $Y_{1},\ldots,Y_{n}$ be independent observations of
$X$ and $Y.$ Let $\left\\{Q_{i}^{X},i=1,\ldots,n\right\\}$ and
$\left\\{Q_{i}^{Y},i=\right.$ $1,\ldots,n\\}$ be the rank of $X_{i}$ and
$Y_{i}$ in the samples $\left\\{X_{i}\right\\}_{i=1}^{n}$ and
$\left\\{Y_{i}\right\\}_{i=1}^{n}.$ Let $\left\\{R_{ni}\right\\}_{i=1}^{n}$
represent the relative ranks:

$R_{ni}=Q_{i^{\prime}}^{Y}\text{ subject to }Q_{i^{\prime}}^{X}=i$

We then have $\left\\{R_{n1},\ldots,R_{nn}\right\\}$ are uniformly distributed
in all permutations of $\\{1,\ldots,n\\}$ with

$\operatorname{pr}\left(R_{n1}=i_{1},\ldots,R_{nn}=i_{n}\right)=\frac{1}{n!}$

for any permeation $\left\\{i_{1},\ldots,i_{n}\right\\}$ of
$\\{1,\ldots,n\\}.$ Here n! represents the factorial of n.

######  Lemma 2.

(Concentration Inequality For Simple Linear Rank Statistics) . Assume the
setting and notation in Lemma 1 . Consider the simple linear rank statistic

$V\equiv\sum_{i=1}^{n}c_{ni}g\left(\frac{R_{ni}}{n+1}\right)=\frac{1}{n}\sum_{i=1}^{n}f\left(\frac{Q_{i}^{X}}{n+1}\right)g\left(\frac{Q_{i}^{Y}}{n+1}\right)$

where $f(\cdot)$ and $g(\cdot)$ are Lipschitz functions with Lipschitz
constant $\Delta<\infty$ and $\max\\{|f(0)|,|g(0)|\\}\leq$ $A_{2}.$ We have,
for any $t>0$

$\operatorname{pr}(|V-EV|>t)\leq 2\exp\left(-Cnt^{2}\right)$

for some scalar $C$ only depending on $\Delta$ and $A_{2}.$

######  Lemma 3.

Suppose that the regularity conditions in Theorem 1 hold. Under the null
hypothesis $H_{0}$ holds, we have in the region
$x\in\left(0,O\left(n^{1/6-\epsilon}\right)\right)$ for some $\epsilon>0$

$\operatorname{pr}\left[\frac{V_{ij}(k)-E_{H_{0}}\left(V_{ij}(k)\right)}{\left\\{\operatorname{var}_{H_{0}}\left(V_{ij}(k)\right)\right\\}^{1/2}}>x\right]=\\{1-\Phi(x)\\}\left\\{1+O\left(\frac{1+x^{3}}{n^{1/2}}+\frac{x}{n^{1/6}}\right)\right\\}$

####  8.1.1 Proof of Theorem 1

And without loss of generality, we assume that $\sum_{i=1}^{n}c_{ni}=0$. Thus,
$E_{H_{0}}(V_{ij}(k))=0$. Let
$\\{\nu_{i}\\}_{i=1}^{N}=\\{V_{ij}(k)/\sigma_{V}\\}_{1\leq k\leq K,1\leq
i,j\leq p}$. Define $z=(2\log(N)-\log\log(N)+y)^{1/2}$. By Lemma 3, we have

$\displaystyle P(|\nu_{i}|\geq
z)=2\\{1-\Phi(z)\\}\\{1+o(1)\\}\sim\frac{1}{\sqrt{\pi}}\frac{e^{-y/2}}{N}$

Thus,

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

$\operatorname{as}p\rightarrow\infty.$ Set $D_{N}:=\left\\{1\leq i\leq
N;\left|B_{N,i}\right|<N^{\varsigma}\right\\}.$ By assumption,
$\left|D_{N}\right|/N\rightarrow 1$ as $N\rightarrow\infty$ Easily,

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

Therefore, to prove Theorem 1, it is enough to show

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

as $N\rightarrow\infty$. Define

$\alpha_{t}=\sum^{*}P\left(\left|\nu_{i_{1}}\right|>z,\cdots,\left|\nu_{i_{t}}\right|>z\right)$

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

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

for each $t\geq 1.$ Because $g(F_{j}(\varepsilon_{t+k,j}))$ is bounded by a
constant $C_{g}$, thus all the assumptions in Theorem 1.1 in Zaitsev (1987)
are satisfied. Thus, we have

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

where $(Z_{i_{1}},\cdots,Z_{i_{t}})$ follows a multivariate normal
distribution with mean zero and the same covariance matrix with
$(\nu_{i_{1}},\cdots,\nu_{i_{t}})$. By the proof of Theorem 2 in Feng et al.
(2022a), we have

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

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

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

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

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

Then, by Bonferroni inequality,

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

for any $k\geq 1.$ let $N\rightarrow\infty$, we have

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

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

####  8.1.2 Proof of Theorem 2

By Lemma 2, there exist a constant $c$ such that, for any $t>0$,

$\displaystyle P\left(|\hat{V}_{ij}(k)-E(\hat{V}_{ij}(k))|>t\right)\leq
2e^{-t^{2}/c}.$

Then,

$\displaystyle P\left(\max_{1\leq i,j\leq p,1\leq k\leq
K}|\hat{V}_{ij}(k)-E(\hat{V}_{ij}(k))|>t\right)\leq N2e^{-t^{2}/c}.$

which implies that, with probability at least $1-N^{-1}$,

$\displaystyle\max_{1\leq i,j\leq p,1\leq k\leq
K}|\hat{V}_{ij}(k)-E(\hat{V}_{ij}(k))|\leq\sqrt{3c\log N}.$

So, for large enough $n$, we have

$\displaystyle V_{n}^{2}/\sigma_{V}^{2}$ $\displaystyle=\max_{1\leq i,j\leq
p,1\leq k\leq K}\widehat{V}_{ij}(k)^{2}\geq\left(\max_{1\leq i,j\leq p,1\leq
k\leq K}\left|E(\hat{V}_{ij}(k))\right|-\max_{1\leq i,j\leq p,1\leq k\leq
K}\left|\widehat{V}_{ij}(k)-E(\hat{V}_{ij}(k))\right|\right)^{2}$
$\displaystyle\geq(2+\varrho)\log N$

for some small positive constant $\varrho$. Accordingly, for any given
$q_{\alpha}$, with probability tending to one,

$\displaystyle V_{n}^{2}/\sigma_{V}^{2}>2\log N-\log\log N-q_{\alpha}.$

Then we complete the proof. $\square$

####  8.1.3 Proof of Theorem 3

According to Theorem 3 in Feng et al. (2022b) and Assumption (A2), we can
easily obtain the result. $\square$

###  8.2 Proof of Theorems of Non-Degenerate U-Statistics

######  Lemma 4.

Suppose that $U$ is a $U$-statistic with degree $m$ and bounded kernel
$|h(\cdot)|\leq M$. We then have, for any $t>0$,

$P(|U-EU|>t)\leq 2\exp\left\\{-nt^{2}/\left(2mM^{2}\right)\right\\}.$

######  Lemma 5.

Suppose that the boundedness assumption in Theorem 4 hold. We then have, in a
region $x\in\left(0,o\left(n^{1/6}\right)\right)$

$P\left[\frac{U_{ij}(k)-E\left(U_{ij}(k)\right)}{\left\\{\operatorname{var}\left(U_{ij}(k)\right)\right\\}^{1/2}}>x\right]=\\{1-\Phi(x)\\}\left\\{1+O\left(\frac{1+x^{3}}{n^{1/2}}\right)\right\\}.$

####  8.2.1 Proof of Theorem 4

First, we consider the following U-statistics with bounded and symmetric
kernels, i.e.

$\displaystyle{U}_{ij}(k)=$ $\displaystyle\frac{1}{C_{n-k}^{m}}\sum_{1\leq
t_{1}<t_{2},\cdots,<t_{m}\leq
n-k}h((\varepsilon_{t_{1},i},\varepsilon_{t_{1}+k,j})^{\top},\cdots,(\varepsilon_{t_{m},i},\varepsilon_{t_{m}+k,j})^{\top})$
(8.19)

Here we define $\\{u_{s}\\}_{s=1}^{N}=\\{U_{ij}(k)\\}_{1\leq i,j\leq p,1\leq
k\leq K}$ and
$\\{{\boldsymbol{X}}_{tijk}\\}=\\{(\varepsilon_{t,i},\varepsilon_{t+k,j})^{\top}\\}_{1\leq
t\leq n-k}$. So we rewrite $U_{ij}(k)$ in the following forms

$\displaystyle u_{s}=\frac{1}{C_{n-k_{s}}^{m}}\sum_{1\leq
t_{1}<t_{2},\cdots,<t_{m}\leq
n-k_{s}}h({\boldsymbol{X}}_{t_{1},i_{s}j_{s}k_{s}},\cdots,{\boldsymbol{X}}_{t_{m},i_{s}j_{s}k_{s}})$
(8.20)

Without loss of generality, we assume that $E(u_{s})=0$. By the condition, we
have

$\displaystyle\mu_{q}\doteq$ $\displaystyle
E|h({\boldsymbol{X}}_{t_{1}},\cdots,{\boldsymbol{X}}_{t_{m}})|^{q}<\infty,$
$\displaystyle\psi_{s}(x)=$ $\displaystyle
E(h({\boldsymbol{X}}_{t_{1}},\cdots,{\boldsymbol{X}}_{t_{m}})|{\boldsymbol{X}}_{t_{1}}=x),~{}~{}\sigma_{\psi}^{2}=\mathrm{var}(\psi_{s}({\boldsymbol{X}}_{t_{1}}))>0.$

for any $q\geq 2$. By Lemma 1 in Malevich and Abdalimov (1979), we can rewrite
$u_{s}$ as follow

$\displaystyle
u_{s}=S_{s}+\eta_{s},~{}~{}S_{s}=\frac{m}{n-k_{s}}\sum_{i=1}^{n-k_{s}}\psi_{s}({\boldsymbol{X}}_{i}),~{}~{}\eta_{s}=\sum_{l=2}^{m}C_{m}^{l}u_{s,l}$

where $u_{s,l}$ is a $U$-statistics of the form (8.20) with kernel
$\psi^{(l)}(x_{1},\cdots,x_{l})$ such that

$\displaystyle\mathbf{P}\left\\{\mathbf{E}\left[\psi^{(l)}\left({\boldsymbol{X}}_{1},\cdots,{\boldsymbol{X}}_{l}\right)\mid{\boldsymbol{X}}_{1},\cdots,{\boldsymbol{X}}_{l-1}\right]=0\right\\}=1,$
$\displaystyle\mathbf{E}\left|\psi^{(l)}\left({\boldsymbol{X}}_{1},\cdots,{\boldsymbol{X}}_{l}\right)\right|^{q}\leqq
2^{mq}\mu_{q}.$

By Lemma 2 in Malevich and Abdalimov (1979), we have

$\displaystyle E|C_{n-k_{s}}^{l}u_{s,l}|^{q}\leq C(n-k_{s})^{lq/2}.$

Thus, by the Markov inequality,

$\displaystyle P(\max_{1\leq s\leq N}(n-k_{s})^{1/2}|\eta_{s}|>(\log N)^{-1})$
$\displaystyle\leq NP(|\eta_{s}|>(\log N)^{-1}(n-k_{s})^{-1/2})$
$\displaystyle\leq N(\log N)^{2q}(n-k_{s})^{q}E(|\eta_{s}|^{2q})$
$\displaystyle\leq N(\log
N)^{2q}(n-k_{s})^{q}E\left(\left|\sum_{l=2}^{m}C_{m}^{l}u_{s,l}\right|^{2q}\right)$
$\displaystyle=N(\log
N)^{2q}(n-k_{s})^{q}m^{2q}E\left(\left|\frac{1}{m}\sum_{l=2}^{m}C_{m}^{l}u_{s,l}\right|^{2q}\right)$
$\displaystyle\leq N(\log
N)^{2q}(n-k_{s})^{q}m^{2q-1}\left\\{\sum_{l=2}^{m}(C_{m}^{l})^{2q}E(|u_{s,l}|^{2q})\right\\}$
$\displaystyle\leq CN(\log
N)^{2q}(n-k_{s})^{q}m^{2q-1}\left\\{\sum_{l=2}^{m}(C_{m}^{l})^{2q}(n-k_{s})^{-lq}\right\\}$
$\displaystyle\leq
Cm^{2q-1}\left\\{\sum_{l=2}^{m}(C_{m}^{l})^{2q}\right\\}N(\log
N)^{2q}(n-k_{s})^{q}(n-k_{s})^{-2q}$ $\displaystyle=C_{0}(m,q)N(\log
N)^{2q}(n-k_{s})^{-q}\to 0,$

for some positive integer $q$ by $N=o(n^{\epsilon})$. Thus, by

$\displaystyle\left|\max_{1\leq s\leq N}(n-k_{s})u_{s}^{2}-\max_{1\leq s\leq
N}(n-k_{s})S_{s}^{2}\right|\leq 2\max_{1\leq s\leq
N}(n-k_{s})^{1/2}|S_{s}|\max_{1\leq s\leq
N}(n-k_{s})^{1/2}|\eta_{s}|+\max_{1\leq s\leq N}(n-k_{s})\eta_{s}^{2}$

we only need to show that

$\operatorname{P}\left(\max_{1\leq s\leq
N}(n-k_{s})S^{2}_{s}/\sigma_{U}^{2}-2\log(Kp^{2})+\log\log(Kp^{2})\leqslant
y\right)\to\exp\left\\{-\pi^{-1/2}\exp(-y/2)\right\\}$

Here we define $\upsilon_{s}=(n-k_{s})^{1/2}S_{s}/\sigma_{U}$ and
$z=(2\log(N)-\log\log(N)+y)^{1/2}$. Since $(n-k_{s})^{1/2}\eta_{s}$ is
negligible, the tail behavior of $\upsilon_{s}$ is the same as that of
$(n-k_{s})^{1/2}u_{s}/\sigma_{U}$. Therefore, by Lemma 5, we have

$\displaystyle P(|\upsilon_{i}|\geq
z)=2\\{1-\Phi(z)\\}\\{1+o(1)\\}\sim\frac{1}{\sqrt{\pi}}\frac{e^{-y/2}}{N}$

Thus,

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

$\operatorname{as}p\rightarrow\infty.$ Set $D_{N}:=\left\\{1\leq i\leq
N;\left|B_{N,i}\right|<N^{\varsigma}\right\\}.$ By assumption,
$\left|D_{N}\right|/N\rightarrow 1$ as $N\rightarrow\infty$ Easily,

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

Therefore, to prove Theorem 4, it is enough to show

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

as $N\rightarrow\infty$. Define

$\alpha_{t}=\sum^{*}P\left(\left|\upsilon_{i_{1}}\right|>z,\cdots,\left|\upsilon_{i_{t}}\right|>z\right)$

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

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

for each $t\geq 1.$ Because $\psi_{s}({\boldsymbol{X}}_{i})$ is bounded, thus
all the assumptions in Theorem 1.1 in Zaitsev (1987) are satisfied. Thus, we
have

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

where $(Z_{i_{1}},\cdots,Z_{i_{t}})$ follows a multivariate normal
distribution with mean zero and the same covariance matrix with
$(\upsilon_{i_{1}},\cdots,\upsilon_{i_{t}})$. By the proof of Theorem 2 in
Feng et al. (2022a), we have

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

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

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

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

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

Then, by Bonferroni inequality,

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

for any $k\geq 1.$ let $N\rightarrow\infty$, we have

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

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

####  8.2.2 Proof of Theorem 5

By Lemma 4, we have

$\displaystyle P\left(|\hat{U}_{ij}(k)-E(\hat{U}_{ij}(k))|>t\right)\leq
2e^{t^{2}/c}$

for some positive constant $c$. Taking the same procedure as Theorem 2, we can
also obtain the result. $\square$

####  8.2.3 Proof of Theorem 6

According to Theorem 3 in Feng et al. (2022b) and Assumption (A3), we can
easily obtain the result. $\square$

###  8.3 Proof of Theorems of Degenerate U-Statistics

####  8.3.1 Proof of Theorem 7

######  Lemma 6.

For $N\geq 1$, let $\iota_{N}$ be positive integers with
$\lim_{N\to\infty}\iota_{N}/N=1$. Let
$Y_{iv},i=1,\cdots,\iota_{N},v=1,\cdots,M$ be $N(0,1)$-distributed random
variables and $\mathrm{cov}(Y_{iv},Y_{is})=0$ for $v\not=s$. Let
${\boldsymbol{Y}}_{i}=(Y_{i1},\cdots,Y_{iM})^{\top}$ and
$\Xi_{ij}=\mathrm{cov}({\boldsymbol{Y}}_{i},{\boldsymbol{Y}}_{j})$. Assume
$|\lambda_{max}(\Xi_{ij}\Xi_{ij}^{\top})|\leq\delta_{N}^{2+2c}$ for $c>0$ and
all $1\leq i<j\leq\iota_{N}$ , where $\\{\delta_{N};\,N\geq 1\\}$ are
constants satisfying $0<\delta_{N}=o(1/\log N).$ Define
$W_{i_{k}}=\sum_{v=1}^{M}\lambda_{v}Y^{2}_{i_{k}v}$. Given $x\in\mathbb{R}$,
set
$z=2\lambda_{1}\log(N)+\lambda_{1}\left(\mu_{1}-2\right)\log\log(N)+\lambda_{1}y+o(1/\log(N)).$
Then, for any fixed $m\geq 1$, we have

$\displaystyle\Big{(}\frac{\Gamma\left(\mu_{1}/2\right)}{\kappa}\frac{N}{e^{-y/2}}\Big{)}^{m}\cdot
P(W_{i_{1}}>z,\cdots,W_{i_{m}}>z)\to 1$ (8.21)

as $N\to\infty$ uniformly for all $1\leq i_{1}<\cdots<i_{m}\leq\iota_{N}.$

Proof of Lemma 6 For $m=1$, (8.21) is followed by Equation (6) in Zolotarev
(1962). Assume Equation (8.21) holds with $m=k-1$. We will prove it also holds
with $m=k$.

Define
${\boldsymbol{Y}}_{S}=({\boldsymbol{Y}}_{i_{1}}^{\top},\cdots,{\boldsymbol{Y}}_{i_{m-1}}^{\top})^{\top}$,
${\boldsymbol{\Sigma}}_{i_{m}S}=\mathrm{cov}({\boldsymbol{Y}}_{i_{m}},{\boldsymbol{Y}}_{S})$
and ${\boldsymbol{\Sigma}}_{Si_{m}}={\boldsymbol{\Sigma}}_{i_{m}S}^{\top}$. So
${\boldsymbol{Y}}_{i_{m}}=({\boldsymbol{Y}}_{i_{m}}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S})+{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\doteq
U_{Y}+V_{Y}$ . Thus, by the conditional distribution of multivariate normal
distributions, we have
${\boldsymbol{Y}}_{i_{m}}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\sim
N(\boldsymbol{0},{\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})$ is
independent of ${\boldsymbol{Y}}_{S}$. Define ${\bf
A}=\mathrm{diag}\\{\lambda_{1},\cdots,\lambda_{M}\\}$. Thus, we have

$\displaystyle P(W_{i_{1}}>z,\cdots,W_{i_{m}}>z)$ $\displaystyle=$
$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\geq z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}+2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\geq
z,2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\leq
C\delta_{N},\min_{1\leq l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf
A}{\boldsymbol{Y}}_{i_{l}}\geq z\right)$
$\displaystyle+P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\geq z,2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}>C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle\leq$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}\geq z-C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)+P\left(2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}>C\delta_{N}\right)$ $\displaystyle\leq$ $\displaystyle
P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq z-C\delta_{N}\right)P\left(\min_{1\leq
l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)+P\left(U_{Y}^{\top}{\bf
A}V_{Y}>C\delta_{N}/4\right)+P\left(V_{Y}^{\top}{\bf
A}V_{Y}>C\delta_{N}/2\right)$

By $U_{Y}\sim N(\boldsymbol{0},{\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})$, we have

$U_{Y}^{\top}{\bf A}U_{Y}\sim\xi_{U}^{\top}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}\xi_{U}$

where $\xi_{U}\sim N(\boldsymbol{0},{\bf I}_{M})$. Define the eigenvalues of
$({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}$
are
$\tilde{\lambda}_{1}\geq\tilde{\lambda}_{2}\geq\cdots\geq\tilde{\lambda}_{M}$
and $\tilde{\Lambda},\tilde{\kappa},\tilde{\mu}_{1}$ are the corresponding
parameters as in Proposition 3.2 in Drton et al. (2020). Because
$\lambda_{max}({\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})\leq\delta_{N}^{2+2c}$,
$\tilde{\lambda}_{1}=\lambda_{1}(1+o(\delta_{N}))$. So does
$\tilde{\Lambda},\tilde{\kappa},\tilde{\mu}_{1}$. So by Equation (6) in
Zolotarev (1962), we have

$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq z-C\delta_{N}\right)$
$\displaystyle\to\frac{\tilde{\kappa}}{\Gamma(\tilde{\mu}_{1}/2)}\left(\frac{z-C\delta_{N}+\tilde{\Lambda}}{2\tilde{\lambda}_{1}}\right)^{\tilde{\mu}_{1}/2-1}\exp\left(-\frac{z-C\delta_{N}+\tilde{\Lambda}}{2\tilde{\lambda}_{1}}\right)$
$\displaystyle\to\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}$

by the assumption $\delta_{N}=o(1/\log N)$. So by the

$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq
z-C\delta_{N}\right)P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)\to\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}\right)^{m}$

Next, we will show that $P\left(U_{Y}^{\top}{\bf
A}V_{Y}>C\delta_{N}/4\right)+P\left(V_{Y}^{\top}{\bf
A}V_{Y}>C\delta_{N}/2\right)=o(N^{-m})$. Similarly,

$\displaystyle V_{Y}^{\top}{\bf
A}V_{Y}\sim\xi_{V}^{\top}{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}\xi_{V}$

where $\xi_{V}\sim N(\boldsymbol{0},{\bf I}_{(m-1)M)}$. Define the eigenvalues
of ${\boldsymbol{\Sigma}}_{Si_{m}}{\bf A}{\boldsymbol{\Sigma}}_{i_{m}S}$ are
$\zeta_{1},\cdots,\zeta_{M}$. So

$\displaystyle V_{Y}^{\top}{\bf
A}V_{Y}\sim\sum_{k=1}^{(m-1)M}\zeta_{k}\xi_{k}^{2}$

where $\xi_{k}$ are all independently distributed as $N(0,1)$. Thus, for small
enough constant $\varpi$,

$\displaystyle E(\exp(\varpi\delta_{N}^{-1-c}V_{Y}^{\top}{\bf A}V_{Y}))=$
$\displaystyle\prod_{k=1}^{M}Ee^{\varpi\delta_{N}^{-1-c}\zeta_{k}\xi_{k}^{2}}=\exp\left\\{-\frac{1}{2}\sum_{k=1}^{(m-1)M}\log[1-2\varpi\delta_{N}^{-1-c}\zeta_{k}]\right\\}$
$\displaystyle\leq$
$\displaystyle\exp\left(2\varpi\delta_{N}^{-1-c}\sum_{k=1}^{(m-1)M}\zeta_{k}\right)$

In addition,

$\displaystyle\sum_{k=1}^{(m-1)M}\zeta_{k}=\mathrm{tr}({\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S})\leq\lambda_{max}({\boldsymbol{\Sigma}}_{Si_{m}}{\boldsymbol{\Sigma}}_{i_{m}S})\mathrm{tr}({\bf
A})\leq\delta_{N}^{2+2c}\Lambda.$

So $E(\exp(\varpi\delta_{N}^{-1-c}V_{Y}^{\top}{\bf
A}V_{Y}))\leq\exp\left(2\varpi\delta_{N}^{1+c}\Lambda\right)\leq C_{2}$ for
some constant $C_{2}>0$. By the Markov inequality, we have

$\displaystyle P\left(V_{Y}^{\top}{\bf A}V_{Y}>C\delta_{N}/2\right)$
$\displaystyle=P\left(\varpi\delta_{N}^{-1-c}V_{Y}^{\top}{\bf
A}V_{Y}>C\varpi\delta_{N}^{-c}/2\right)$
$\displaystyle\leq\exp\left(-C\varpi\delta_{N}^{-c}/2\right)E(\exp(\varpi\delta_{N}^{-1-c}V_{Y}^{\top}{\bf
A}V_{Y}))$ $\displaystyle\leq
C_{2}\exp\left(-C\varpi\delta_{N}^{-c}/2\right)=o(N^{-m})$

for large enough constant $C$. Similarly,

$\displaystyle U_{Y}^{\top}{\bf
A}V_{Y}\sim\sum_{k=1}^{M}\rho_{k}\xi_{k}\eta_{k}$

where $\eta_{k}$ are all independently distributed as $N(0,1)$,
$\rho_{1},\cdots,\rho_{M}$ are the singular values of $({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}$. And then

$\displaystyle P\left(U_{Y}^{\top}{\bf A}V_{Y}>C\delta_{N}/4\right)$
$\displaystyle=$ $\displaystyle P\left(\varpi\delta_{N}^{-1-c}U_{Y}^{\top}{\bf
A}V_{Y}>C\varpi\delta_{N}^{-c}/4\right)$ $\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)E(\exp(\varpi\delta_{N}^{-1-c}U_{Y}^{\top}{\bf
A}V_{Y}))$ $\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)\exp\left(-\frac{1}{2}\sum_{k=1}^{M}\log[1-\varpi^{2}\delta_{N}^{-2-2c}\rho_{k}^{2}]\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)\exp\left(\varpi^{2}\delta_{N}^{-2-2c}\sum_{k=1}^{M}\rho_{k}^{2}\right)$
$\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)\exp\left(\varpi^{2}\delta_{N}^{-2-2c}\mathrm{tr}((({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S})^{2})\right)$ $\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)\exp\left(\varpi^{2}\delta_{N}^{-2-2c}\lambda_{max}({\boldsymbol{\Sigma}}_{Si_{m}}{\boldsymbol{\Sigma}}_{i_{m}S})\mathrm{tr}({\bf
A}^{2})\right)$ $\displaystyle\leq$
$\displaystyle\exp\left(-C\varpi\delta_{N}^{-c}/4\right)\exp\left(\varpi^{2}\mathrm{tr}({\bf
A}^{2})\right)=o(N^{-m})$

for large enough constant $C>0$. Thus, we have

$P(W_{i_{1}}>z,\cdots,W_{i_{m}}>z)\leq\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}\right)^{m}+o(N^{-m}).$
(8.22)

Further more,

$\displaystyle P(W_{i_{1}}>z,\cdots,W_{i_{m}}>z)$ $\displaystyle=$
$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\geq z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle\geq$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}\geq z+C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)-P\left(2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}\leq-C\delta_{N}\right)$ $\displaystyle\geq$ $\displaystyle
P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq z+C\delta_{N}\right)P\left(\min_{1\leq
l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)-P\left(2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}\leq-C\delta_{N}\right),$

where the first inequality is based on the fact that:

$\displaystyle P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)-P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}\geq z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}+2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}<z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}+2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}<z,2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}<-C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle+P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}<z,2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}\geq-C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle\leq$ $\displaystyle P\left(2U_{Y}^{\top}{\bf
A}V_{Y}+V_{Y}^{\top}{\bf A}V_{Y}<-C\delta_{N}\right)+P\left(U_{Y}^{\top}{\bf
A}U_{Y}<z+C\delta_{N},\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right).$

Obviously,

$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq
z+C\delta_{N}\right)P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)\to\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}\right)^{m}$

by $\delta_{N}=o(1/\log N)$. Next,

$\displaystyle P\left(2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}\leq-C\delta_{N}\right)$ $\displaystyle\leq$ $\displaystyle
P\left(U_{Y}^{\top}{\bf
A}V_{Y}\leq-C\delta_{N}/2\right)=P\left(U_{Y}^{\top}{\bf A}V_{Y}\geq
C\delta_{N}/2\right)=o(N^{-m}).$

So

$\displaystyle
P(W_{i_{1}}>z,\cdots,W_{i_{m}}>z)\geq\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}\right)^{m}+o(N^{-m}).$
(8.23)

Then, we obtain the result by (8.22) and (8.23). $\square$

######  Lemma 7.

Let $W_{i}=\sum_{v=1}^{M}\lambda_{v}Y^{2}_{iv}$ where
$Y_{iv}\stackrel{{\scriptstyle i.i.d}}{{\sim}}N(0,1)$ for $v=1,\cdots,M$.
Define ${\boldsymbol{Y}}_{i}=(Y_{i1},\cdots,Y_{iM})^{\top}$ and
${\boldsymbol{Y}}_{S}=({\boldsymbol{Y}}_{i_{1}}^{\top},\cdots,{\boldsymbol{Y}}_{i_{m-1}}^{\top})^{\top}$
where $S=\\{i_{1},\cdots,i_{m-1}\\}$ which for any $i,j\in S$,
$\lambda_{max}(\Xi_{ij}\Xi_{ij}^{\top})\leq\delta_{N}^{2+2c_{0}}$ for some
constant $c_{0}>0$. Let
$\Xi_{ij}=\mathrm{cov}({\boldsymbol{Y}}_{i},{\boldsymbol{Y}}_{j})$. Let
${\boldsymbol{Y}}_{i_{m}}$ satisfy $\max_{1\leq j\leq
m-1}|\lambda_{max}(\Xi_{i_{m}j}\Xi_{i_{m}j}^{\top})|>\delta_{N}^{2+2c_{0}}$
and
$\lambda_{max}({\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})\leq\delta\in(0,1)$
where
${\boldsymbol{\Sigma}}_{i_{m}S}=\mathrm{cov}({\boldsymbol{Y}}_{i_{m}},{\boldsymbol{Y}}_{S})$
and ${\boldsymbol{\Sigma}}_{Si_{m}}={\boldsymbol{\Sigma}}_{i_{m}S}^{\top}$.
Then, we have

$\displaystyle P\left(W_{i_{m}}\geq z,\min_{1\leq l\leq m-1}W_{i_{l}}\geq
z\right)\leq C(\log N)^{c}N^{\varrho-m}$ (8.24)

where $\varrho\in(0,1)$, $c,C$ are some positive constants.

Proof. Define
${\boldsymbol{Y}}_{i_{m}}=({\boldsymbol{Y}}_{i_{m}}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S})+{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\doteq
U_{Y}+V_{Y}$ . Thus, by the conditional distribution of multivariate normal
distributions, we have
${\boldsymbol{Y}}_{i_{m}}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\sim
N(\boldsymbol{0},{\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})$ is
independent of ${\boldsymbol{Y}}_{S}$. Define ${\bf
A}=\mathrm{diag}\\{\lambda_{1},\cdots,\lambda_{M}\\}$. Thus, we have

$\displaystyle P\left(W_{i_{m}}\geq z,\min_{1\leq l\leq m-1}W_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle
P\left({\boldsymbol{Y}}_{i_{m}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{m}}\geq
z,\min_{1\leq l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf
A}{\boldsymbol{Y}}_{i_{l}}\geq z\right)$ $\displaystyle=$ $\displaystyle
P\left(U_{Y}^{\top}{\bf A}U_{Y}+2U_{Y}^{\top}{\bf A}V_{Y}+V_{Y}^{\top}{\bf
A}V_{Y}\geq z,\min_{1\leq l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf
A}{\boldsymbol{Y}}_{i_{l}}\geq z\right)$ $\displaystyle\leq$ $\displaystyle
P\left(2U_{Y}^{\top}{\bf A}U_{Y}+2V_{Y}^{\top}{\bf A}V_{Y}\geq z,\min_{1\leq
l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle\leq$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}\geq\frac{1}{4}z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)+P\left(V_{Y}^{\top}{\bf A}V_{Y}\geq\frac{1}{4}z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}\geq\frac{1}{4}z\right)P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)+P\left(V_{Y}^{\top}{\bf A}V_{Y}\geq\frac{1}{4}z,\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$

By Lemma 6, we have

$\displaystyle P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)\leq(1+2\epsilon)^{m-1}\Big{(}\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}\frac{e^{-y/2}}{N}\Big{)}^{m-1}\leq
CN^{1-m}$

By $U_{Y}\sim N(\boldsymbol{0},{\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})$, we have

$U_{Y}^{\top}{\bf A}U_{Y}\sim\xi_{U}^{\top}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}\xi_{U}$

where $\xi_{U}\sim N(\boldsymbol{0},{\bf I}_{M})$. So

$\displaystyle\tilde{\lambda}_{1}$ $\displaystyle\doteq\lambda_{max}((({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2}{\bf
A}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})^{1/2})$
$\displaystyle\geq\lambda_{max}({\bf A})\lambda_{\min}({\bf
I}_{M}-{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{\Sigma}}_{Si_{m}})\geq\lambda_{1}(1-\delta)$

by the assumption. By Equation (6) in Zolotarev (1962), we have

$\displaystyle P\left(U_{Y}^{\top}{\bf A}U_{Y}\geq\frac{1}{4}z\right)\leq$
$\displaystyle(1+\epsilon)\frac{\tilde{\kappa}}{\Gamma(\tilde{\mu}_{1}/2)}\left(\frac{z/4+\tilde{\Lambda}}{2\tilde{\lambda}_{1}}\right)^{\tilde{\mu}_{1}/2-1}\exp\left(-\frac{z/4+\tilde{\Lambda}}{2\tilde{\lambda}_{1}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{z/4+\tilde{\Lambda}}{2\lambda_{1}(1-\delta)}\right)^{\tilde{\mu}_{1}/2-1}\exp\left(-\frac{z/4+\tilde{\Lambda}}{2\lambda_{1}(1-\delta)}\right)$
$\displaystyle\leq$ $\displaystyle C(\log N)^{c}N^{-(1-\delta)/4}$

where $c=(\tilde{\mu}_{1}-\mu_{1})/2$. Thus, we have

$\displaystyle P\left(U_{Y}^{\top}{\bf
A}U_{Y}\geq\frac{1}{4}z\right)P\left(\min_{1\leq l\leq
m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)\leq C(\log N)^{c}N^{-m+(3+\delta)/4}$ (8.25)

Define $\tilde{\bf A}=\mathrm{diag}\\{{\bf A},\cdots,{\bf
A}\\}\in\mathbb{R}^{(m-1)M\times(m-1)M}$. Next, we consider

$\displaystyle P\left(V_{Y}^{\top}{\bf A}V_{Y}\geq\frac{1}{4}z,\min_{1\leq
l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle=$ $\displaystyle
P\left({\boldsymbol{Y}}_{S}^{\top}{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\geq\frac{1}{4}z,\min_{1\leq
l\leq m-1}{\boldsymbol{Y}}_{i_{l}}^{\top}{\bf A}{\boldsymbol{Y}}_{i_{l}}\geq
z\right)$ $\displaystyle\leq$ $\displaystyle
P\left({\boldsymbol{Y}}_{S}^{\top}{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}{\boldsymbol{Y}}_{S}\geq\frac{1}{4}z,{\boldsymbol{Y}}_{S}^{\top}\tilde{\bf
A}{\boldsymbol{Y}}_{S}\geq(m-1)z\right)$ $\displaystyle\leq$ $\displaystyle
P\left({\boldsymbol{Y}}_{S}^{\top}((1-\epsilon)\tilde{\bf
A}+\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}){\boldsymbol{Y}}_{S}\geq((m-1)(1-\epsilon)+\frac{1}{4}\epsilon)z\right)$
$\displaystyle=$ $\displaystyle
P\left(\xi_{S}^{\top}{\boldsymbol{\Sigma}}_{S}^{1/2}((1-\epsilon)\tilde{\bf
A}+\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}){\boldsymbol{\Sigma}}_{S}^{1/2}\xi_{S}\geq((1-\epsilon)m-1+\frac{5}{4}\epsilon)z\right)$

where $\xi_{S}\sim N(\boldsymbol{0},{\bf I}_{(m-1)M})$. We have

$\displaystyle\breve{\lambda}_{1}\doteq$
$\displaystyle\lambda_{max}\left({\boldsymbol{\Sigma}}_{S}^{1/2}((1-\epsilon)\tilde{\bf
A}+\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}){\boldsymbol{\Sigma}}_{S}^{1/2}\right)$
$\displaystyle\geq$
$\displaystyle(1-(m-1)\delta_{N})\lambda_{max}\left((1-\epsilon)\tilde{\bf
A}+\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}\right)$ $\displaystyle\geq$
$\displaystyle(1-(m-1)\delta_{N})\left(\lambda_{max}\left((1-\epsilon)\tilde{\bf
A}\right)-\lambda_{max}\left(\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}\right)\right)$ $\displaystyle\geq$
$\displaystyle\lambda_{1}(1-(m-1)\delta_{N})(1-(1+\delta)\epsilon)\geq(1-\delta/2)\lambda_{1}$

for a small enough positive real number $\epsilon$.

By Equation (6) in Zolotarev (1962), we have

$\displaystyle
P\left(\xi_{S}^{\top}{\boldsymbol{\Sigma}}_{S}^{1/2}((1-\epsilon)\tilde{\bf
A}+\epsilon{\boldsymbol{\Sigma}}_{Si_{m}}{\bf
A}{\boldsymbol{\Sigma}}_{i_{m}S}){\boldsymbol{\Sigma}}_{S}^{1/2}\xi_{S}\geq((1-\epsilon)m-1+\frac{5}{4}\epsilon)z\right)$
$\displaystyle\leq$
$\displaystyle(1+\epsilon)\frac{{\breve{\kappa}}}{\Gamma(\breve{\mu}_{1}/2)}\left(\frac{((1-\epsilon)m-1+\frac{5}{4}\epsilon)z+\breve{\Lambda}}{2\breve{\lambda}_{1}}\right)^{\breve{\mu}_{1}/2-1}\exp\left(-\frac{((1-\epsilon)m-1+\frac{5}{4}\epsilon)z+\breve{\Lambda}}{2\breve{\lambda}_{1}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{((1-\epsilon)m-1+\frac{5}{4}\epsilon)z+\breve{\Lambda}}{2\lambda_{1}(1-\delta/2)}\right)^{\breve{\mu}_{1}/2-1}\exp\left(-\frac{((1-\epsilon)m-1+\frac{5}{4}\epsilon)z+\breve{\Lambda}}{2\lambda_{1}(1-\delta/2)}\right)$
$\displaystyle\leq$ $\displaystyle C(\log
N)^{c}N^{-\frac{(1-\epsilon)m-1+\frac{5}{4}\epsilon}{1-\delta/2}}\leq C(\log
N)^{c}N^{-m+(3+\delta)/4}$

by setting $\frac{(1-\epsilon)m-1+\frac{5}{4}\epsilon}{1-\delta/2}\geq
m-(3+\delta)/4$. $\square$

######  Lemma 8.

Define
$\upsilon_{t}=\sum_{v=1}^{M}\lambda_{v}\left({(n-k_{t})}^{-1/2}\sum_{i=1}^{{n-k_{t}}}\phi_{v}(Z_{i,t})\right)^{2}$
where $Z_{i,t}$ is the corresponding random variable in Condition (C6) with
respect to $t\in\\{1,\cdots,N\\}$. Let

$\beta_{s}=\sum^{*}P\left(\upsilon_{i_{1}}>z,\cdots,\upsilon_{i_{s}}>z\right)$

for $1\leq s\leq N,$ where the sum runs over all $i_{1}<\cdots<i_{s}$ and
$i_{1}\in D_{N},\cdots,i_{s}\in D_{N}.$ Then,

$\lim_{N\rightarrow\infty}\beta_{s}=\frac{1}{s!}\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}e^{-\frac{y}{2}}\right)^{-s}$

for each $s\geq 1.$

Proof. According to Theorem 1.1 in Zaitsev (1987), we have

$\displaystyle\sum^{*}P\left(\sum_{v=1}^{M}\lambda_{v}Y_{i_{1}v}^{2}>z+\epsilon_{n}(\log(N))^{-1},\cdots,\sum_{v=1}^{M}\lambda_{v}Y_{i_{s}v}^{2}>z+\epsilon_{n}(\log(N))^{-1}\right)$
$\displaystyle-\left(\begin{array}[]{c}|D_{N}|\\\
s\end{array}\right)c_{1}s^{5/2}\exp\left(-\frac{n^{1/2}\epsilon_{n}}{c_{2}s^{3}(\log
N)}\right)$ $\displaystyle\leq$
$\displaystyle\sum^{*}P\left(\upsilon_{i_{1}}>z,\cdots,\upsilon_{i_{s}}>z\right)$
$\displaystyle\leq$
$\displaystyle\sum^{*}P\left(\sum_{v=1}^{M}\lambda_{v}Y_{i_{1}v}^{2}>z-\epsilon_{n}(\log(N))^{-1},\cdots,\sum_{v=1}^{M}\lambda_{v}Y_{i_{s}v}^{2}>z-\epsilon_{n}(\log(N))^{-1}\right)$
$\displaystyle+\left(\begin{array}[]{c}|D_{N}|\\\
s\end{array}\right)c_{1}s^{5/2}\exp\left(-\frac{n^{1/2}\epsilon_{n}}{c_{2}s^{3}(\log
N)}\right)$

where $(Y_{i_{1}v},\cdots,Y_{i_{s}v})$ follows a multivariate normal
distribution with mean zero and the same covariance matrix with
$({(n-k_{i_{1}})}^{-1/2}\sum_{j=1}^{{n-k_{i_{1}}}}\phi_{v}(Z_{j,{i_{1}}}),\cdots,{(n-k_{i_{s}})}^{-1/2}\sum_{j=1}^{{n-k_{i_{s}}}}\phi_{v}(Z_{j,{i_{s}}}))$.
By the condition $N=o(n^{\epsilon})$, there exist small enough
$\epsilon_{n}\to 0$ satisfy

$\left(\begin{array}[]{c}|D_{N}|\\\
s\end{array}\right)c_{1}s^{5/2}\exp\left(-\frac{n^{1/2}\epsilon_{n}}{c_{2}s^{3}(\log
N)}\right)\to 0.$

Define $W_{i}=\sum_{v=1}^{M}\lambda_{v}Y_{iv}^{2}$ and
${\boldsymbol{Y}}_{i}=(Y_{i1},\cdots,Y_{iM})$. So we only need to show that

$\sum^{*}P\left(W_{i_{1}}>z,\cdots,W_{i_{s}}>z\right)\to\frac{1}{s!}\left(\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}e^{-\frac{y}{2}}\right)^{-s}.$

Recalling $D_{N}:=\\{1\leq i\leq N;\,|B_{N,i}|<N^{\kappa}\\}$, we write

$\displaystyle\big{\\{}(i_{1},\cdots,i_{t})\in(D_{N})^{t};\,i_{1}<\cdots<i_{t}\big{\\}}=F_{t}\cup
G_{t}$

where
$\sigma_{i_{r}i_{s}}=\lambda_{max}(\Xi_{i_{r}i_{s}}\Xi_{i_{r}i_{s}}^{\top})$
and
$\Xi_{i_{r}i_{s}}=\mathrm{cov}({\boldsymbol{Y}}_{i_{r}},{\boldsymbol{Y}}_{i_{s}})$.

$\displaystyle
F_{t}:=\big{\\{}(i_{1},\cdots,i_{t})\in(D_{N})^{t};\,i_{1}<\cdots<i_{t}\
\mbox{and}\ |\sigma_{i_{r}i_{s}}|\leq\delta_{N}^{2+2c}\ \mbox{for all}\ 1\leq
r<s\leq t\\};$ $\displaystyle
G_{t}:=\big{\\{}(i_{1},\cdots,i_{t})\in(D_{N})^{t};\,i_{1}<\cdots<i_{t}\
\mbox{and}\ |\sigma_{i_{r}i_{s}}|>\delta_{N}^{2+2c}\ \mbox{for a pair}\
(i_{r},i_{s})\ \mbox{with}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1\leq
r<s\leq t\big{\\}}.$ (8.26)

Now, think $D_{N}$ as graph with $|D_{N}|$ vertices. Keep in mind that
$|D_{N}|\leq N$ and $|D_{N}|/N\to 1$. Any two different vertices from them,
say, $i$ and $j$ are connected if $|\sigma_{ij}|>\delta_{N}^{2+2c}.$ In this
case we also say there is an edge between them. By the definition $D_{N}$,
each vertex in the graph has at most $N^{\varsigma}$ neighbors. Replacing
“$n$”, “$q$” and “$t$” in Lemma 7.1 in Feng et al. (2022a) with “$|D_{N}|$”,
“$N^{\varsigma}$” and “$t$”, respectively, we have that $|G_{t}|\leq
N^{t+\varsigma-1}$ for each $2\leq t\leq N$. Therefore
$\binom{|D_{N}|}{t}\geq|F_{t}|\geq\binom{|D_{N}|}{t}-N^{t+\varsigma-1}$. Since
$D_{N}/N\to 1$ and $\varsigma=\varsigma_{N}\to 0$ as $N\to\infty$, we know

$\displaystyle\lim_{N\to\infty}\frac{|F_{t}|}{N^{t}}=\frac{1}{t!}.$ (8.27)

Here

$\displaystyle\beta_{t}$ $\displaystyle=$
$\displaystyle\sum_{(i_{1},\cdots,i_{t})\in
F_{t}}P(W_{i_{1}}>z,\cdots,W_{i_{t}}>z)+$
$\displaystyle\sum_{(i_{1},\cdots,i_{t})\in
G_{t}}P(W_{i_{1}}>z,\cdots,W_{i_{t}}>z).$

From Lemma 6 and (8.27) we have

$\displaystyle\sum_{(i_{1},\cdots,i_{t})\in
F_{t}}P(W_{i_{1}}>z,\cdots,W_{i_{t}}>z)\to\frac{1}{t!}\Big{(}\frac{\kappa}{\Gamma\left(\mu_{1}/2\right)}e^{-\frac{y}{2}}\Big{)}^{t}$

as $N\to\infty.$ As a consequence, it remains to show

$\displaystyle\sum_{(i_{1},\cdots,i_{t})\in
G_{t}}P(W_{i_{1}}>z,\cdots,W_{i_{t}}>z)\to 0$ (8.28)

as $N\to\infty$ for each $t\geq 2.$