# A CLT for the LSS of large dimensional sample covariance matrices with
diverging spikes

Zhijun Liulabel=e1 [ mark]<EMAIL_ADDRESS>Jiang Hulabel=e2 [
<EMAIL_ADDRESS>Zhidong Bailabel=e3 [<EMAIL_ADDRESS>Haiyan
Songlabel=e4 [<EMAIL_ADDRESS>KLASMOE and School of Mathematics
and Statistics, Northeast Normal University, China.  Zhijun Liulabel=e1 [
mark]<EMAIL_ADDRESS>Jiang Hulabel=e2 [<EMAIL_ADDRESS>Zhidong Bailabel=e3 [<EMAIL_ADDRESS>Haiyan Songlabel=e4 [
<EMAIL_ADDRESS>KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China.

# Supplement to “A CLT for the LSS of large dimensional sample covariance
matrices with diverging spikes”

Zhijun Liulabel=e1 [ mark]<EMAIL_ADDRESS>Jiang Hulabel=e2 [
<EMAIL_ADDRESS>Zhidong Bailabel=e3 [<EMAIL_ADDRESS>Haiyan
Songlabel=e4 [<EMAIL_ADDRESS>KLASMOE and School of Mathematics
and Statistics, Northeast Normal University, China.  Zhijun Liulabel=e1 [
mark]<EMAIL_ADDRESS>Jiang Hulabel=e2 [<EMAIL_ADDRESS>Zhidong Bailabel=e3 [<EMAIL_ADDRESS>Haiyan Songlabel=e4 [
<EMAIL_ADDRESS>KLASMOE and School of Mathematics and Statistics,
Northeast Normal University, China.

###### Abstract

In this paper, we establish the central limit theorem (CLT) for linear
spectral statistics (LSSs) of a large-dimensional sample covariance matrix
when the population covariance matrices are involved with diverging spikes.
This constitutes a nontrivial extension of the Bai-Silverstein theorem (BST)
(Ann Probab 32(1):553–605, 2004), a theorem that has strongly influenced the
development of high-dimensional statistics, especially in the applications of
random matrix theory to statistics. Recently, there has been a growing
realization that the assumption of uniform boundedness of the population
covariance matrices in the BST is not satisfied in some fields, such as
economics, where the variances of principal components may diverge as the
dimension tends to infinity. Therefore, in this paper, we aim to eliminate
this obstacle to applications of the BST. Our new CLT accommodates spiked
eigenvalues, which may either be bounded or tend to infinity. A distinguishing
feature of our result is that the variance in the new CLT is related to both
spiked eigenvalues and bulk eigenvalues, with dominance being determined by
the divergence rate of the largest spiked eigenvalues. The new CLT for LSS is
then applied to test the hypothesis that the population covariance matrix is
the identity matrix or a generalized spiked model. The asymptotic
distributions of the corrected likelihood ratio test statistic and the
corrected Nagao’s trace test statistic are derived under the alternative
hypothesis. Moreover, we present power comparisons between these two LSSs and
Roy’s largest root test. In particular, we demonstrate that except for the
case in which the number of spikes is equal to one, the LSSs could exhibit
higher asymptotic power than Roy’s largest root test.

60B20,

60F05,

Empirical spectral distribution,

linear spectral statistic,

random matrix,

Stieltjes transform,

60B20,

60F05,

Empirical spectral distribution,

linear spectral statistic,

random matrix,

Stieltjes transform,

###### keywords:

[class=MSC]

###### keywords:

###### keywords:

[class=MSC]

###### keywords:

, , , and

##  1 Introduction

We consider the general sample covariance matrix ${\bf B}_{n}=\frac{1}{n}{\bf
T}_{p}{\bf X}_{n}{\bf X}_{n}^{\ast}{\bf T}_{p}^{\ast}$, where ${\bf X}_{n}$ is
a $p\times n$ matrix with independent and identically distributed (i.i.d.)
standardized entries $\left\\{x_{ij}\right\\}_{1\leq i\leq p,1\leq j\leq n}$,
${\bf T}_{p}$ is a $p\times p$ deterministic matrix, ${\bf T}_{p}{\bf X}_{n}$
is considered a random sample from the population with the population
covariance matrix ${\bf T}_{p}{\bf T}_{p}^{\ast}=\boldsymbol{\Sigma}$, and ∗
represents the complex conjugate transpose. In the sequel, we simply write
${\bf B}\equiv{\bf B}_{n}$, ${\bf T}\equiv{\bf T}_{p}$ and ${\bf X}\equiv{\bf
X}_{n}$ when there is no confusion. Let $\lambda_{1}\geq\dots\geq\lambda_{p}$
be the eigenvalues of ${\bf B}$. For a known test function $f$, we call
$\sum_{j=1}^{p}f\left({\bf\lambda}_{j}\right)$ a linear spectral statistic
(LSS) of ${\bf B}$. Because most of the classical test statistics in
multivariate statistical analysis are associated with the eigenvalues of
sample covariance matrices, LSSs are remarkable tools in many statistical
problems (see Anderson (2003); Yao et al. (2015) for details).  Through
extensive study of high-dimensional data, it has been discovered that the
distributions of LSSs significantly differ between low-dimensional and high-
dimensional data. For example, in the low-dimensional setting, Wilks’ theorem
(see Wilks (1938)) provides the $\chi^{2}$ approximation for the likelihood
ratio test (LRT) statistic, which is a kind of LSS. However, when $p$ is large
compared with the sample size $n$, the LRT statistic exhibits Gaussian
fluctuations (see Bai et al. (2009); Jiang and Yang (2013)). More generally,
Bai and Silverstein (2004) established the central limit theorem (CLT) for the
LSSs of a high-dimensional ${\bf B}$ under Gaussian-like moments condition by
employing random matrix theory (RMT). Here the term ‘Gaussian-like moments’
refers to the population second-order and fourth-order moments are the same as
those of real or complex Gaussian population. We refer to this CLT as the
Bai–Silverstein theorem (BST) for brevity. Following the work of Bai and
Silverstein (2004), many extensions have been developed under different
settings. Pan and Zhou (2008) generalized the BST by relaxing the Gaussian-
like moments condition of $x_{ij}$, which at the price of adding a structural
condition on ${\bf T}$.  Zheng (2012), Yang and Pan (2015) and Bao et al.
(2022) extended the BST to multivariate $F$ matrices, canonical correlation
matrices and block correlation matrices, respectively. Pan (2014) presented
the CLT for the LSS of noncentered sample covariance matrices, and Zheng et
al. (2015) studied the case of an unbiased sample covariance matrix when the
population mean is unknown. Chen and Pan (2015) focused on the ultrahigh
dimensional case in which the dimension $p$ is much larger than the sample
size $n$. Gao et al. (2017) and Li et al. (2021) studied the CLTs for the LSSs
of high-dimensional Spearman and Kendall’s rank correlation matrices,
respectively. Without attempting to be comprehensive, we also refer readers to
other extensions (Bai et al., 2007, 2015, 2019; Zheng et al., 2019; Banna et
al., 2020; Najim and Yao, 2016; Baik et al., 2018; Hu et al., 2019; Jiang and
Bai, 2021).

Almost all the literature mentioned above have traditionally assumed that the
spectral norms of $\boldsymbol{\Sigma}$ are bounded in $n$. This assumption
limits the applications in data analysis because in many fields, such as
economics and wireless communication networks, the leading eigenvalues may
tend to infinity. We present two examples here.

  * •

Signal detection (Johnstone and Nadler (2017)): We consider a single signal
model:

${\bm{x}}=\chi_{s}^{1/2}u{\bm{h}}+\sigma\bm{v},$

where ${\bm{h}}$ is an unknown $p$-dimensional unit vector, $u$ is a random
variable distributed as $N(0,1)$, $\chi_{s}$ is the signal strength, $\sigma$
is the noise level, and $\bm{v}$ is a random noise vector that is independent
of $u$ and follows a multivariate Gaussian distribution
$N_{p}(0,\bm{\Sigma}_{v})$. It is easy to check that the covariance matrix of
$\bm{x}$ is
$\bm{\Sigma}_{x}=\sigma^{2}\bm{\Sigma}_{v}+\chi_{s}{\bm{h}}{\bm{h}}^{\top}$.
When the noise level is low, but the signal strength is large and sometimes
tends to infinity, it is illogical to assume the boundedness of
$\bm{\Sigma}_{x}$.

  * •

Factor model (Bai and Ng (2002)): Many economic analyses, such as arbitrage
pricing theory and analyses of the rank of a demand system, align naturally
within the framework of the factor model:

$\displaystyle\mathop{\bm{x}_{t}}\limits_{\left(N\times
1\right)}=\mathop{{\bf\Lambda}}\limits_{\left(N\times
r\right)}\mathop{\bm{f}_{t}}\limits_{\left(r\times
1\right)}+\mathop{\bm{\varepsilon}_{t}}\limits_{\left(N\times
1\right)}~{}~{}t=1,\dots,T.$

where $\bm{x}_{t}$ is the observed data, $N$ represents the number of cross-
sections, $T$ is a large time dimension, and $\bm{f}_{t},{\bf\Lambda}$ and
$\bm{\varepsilon}_{t}$ represent the common factors, the factor loadings and
the idiosyncratic error term, respectively. To ensure the identification of
the model, several conventional assumptions are needed, such as
$\mathbb{E}\bm{f}_{t}=\bf 0$,
$\mathbb{E}\left(\bm{f}_{t}\bm{f}_{t}^{\top}\right)={\bf I}_{r}$,
$\bm{\varepsilon}_{t}$ is independent of $\bm{f}_{t}$ with
$\mathbb{E}\bm{\varepsilon}_{t}=\bf 0$ and
$\mathbb{E}\bm{\varepsilon}_{t}\bm{\varepsilon}_{t}^{\top}=\boldsymbol{\Sigma}_{\varepsilon}>0$.
Then the covariance matrices of $\bm{x}_{t}$ can be expressed as
$\boldsymbol{\Sigma}_{x}={\bf\Lambda}{\bf\Lambda}^{\top}+\boldsymbol{\Sigma}_{\varepsilon}$.
A pervasive assumption is that the variances of the principal components
${\bf\Lambda}\bm{f}_{t}$ can diverge as $N$ increases to infinity (see
Assumption B of Bai and Ng (2002)). Therefore, the spectral norms of
$\boldsymbol{\Sigma}_{x}$ are unbounded.

For these reasons, it is of practical value to obtain the asymptotic
properties of the LSS when $\boldsymbol{\Sigma}$ is unbounded. Therefore, in
this paper, we focus on a generalized CLT for the LSSs of a spiked covariance
matrix with the following structure:

$\displaystyle\boldsymbol{\Sigma}=\mathbf{V}\left(\begin{array}[]{cc}{\bf
D}_{1}&0\\\ 0&{\bf D}_{2}\end{array}\right)\mathbf{V}^{\ast},$ (1.3)

where $\mathbf{V}$ is a unitary matrix, ${\bf D}_{1}$ is a diagonal matrix
consisting of the descending unbounded eigenvalues, and ${\bf D}_{2}$ is the
diagonal matrix of the bounded eigenvalues. As an application, the established
CLT is employed to study the asymptotic behaviors of two special LSSs, i.e.,
the likelihood ratio (LR) statistic and Nagao’s trace (NT) statistic, under
the hypothesis

$\displaystyle H_{0}:\boldsymbol{\Sigma}={\bf I}_{p}\quad\text{vs.}\quad
H_{1}:\boldsymbol{\Sigma}=\mathbf{V}\left(\begin{array}[]{cc}{\bf D}_{1}&0\\\
0&{\bf I}_{p-M}\end{array}\right)\mathbf{V}^{\ast},$ (1.6)

where $M$ is a constant. We also derive the asymptotic power of Roy’s largest
root test to detect the above hypothesis and make a comparison with these two
LSSs.

The setting (1.3) is attributed to the famous spiked model in which a few
large eigenvalues of the population covariance matrix are assumed to be well
separated from the remaining eigenvalues (Johnstone, 2001). The spiked model
has served as the foundation for a rich theory of principal component analysis
through the performance of extreme eigenvalues, as discussed in Baik and
Silverstein (2006); Paul (2007); Bai and Yao (2008); Nadler (2008); Jung and
Marron (2009); Bai and Yao (2012); Onatski et al. (2014); Bloemendal et al.
(2016); Wang and Yao (2017); Donoho et al. (2018); Perry et al. (2018);
Johnstone and Paul (2018); Yang and Johnstone (2018); Yao et al. (2018);
Dobriban (2020); Johnstone and Onatski (2020); Cai et al. (2020); Jiang and
Bai (2021). There are also several works that have considered the asymptotic
behaviors of various quantities as the spike strengths tend to infinity.
Specifically, Zhou and Marron (2015) focused on the consistency of the sample
eigenvector, corresponding to the largest eigenvalue of the sample covariance
matrix, under high dimension and low sample size settings, when
$\boldsymbol{\Sigma}$ is unbounded and the data set is Gaussian. Wang and Fan
(2017) derived the asymptotic distributions of the spiked eigenvalues and
eigenvectors when $\boldsymbol{\Sigma}$ is diagonal and unbounded, and the
data set is sub-Gaussian. Recently, Li et al. (2020), Yin (2021) and Zhang et
al. (2022) investigated the trace of a large sample covariance matrix under
the spiked model assumption.

To summarize, the contributions of this paper are as follows.

  1. 1.

We demonstrate a nontrivial extension of the BST to the situation in which the
spectral norms of the population covariance matrices are allowed to diverge as
$\min\\{p,n\\}\to\infty$. In particular, we show how the test function $f$ and
the divergence rate of the population spectral norm affect the new CLT.

  2. 2.

It was previously reported that Gaussian-like moments or diagonality of the
population covariance matrix are necessary for the CLT of the LSS (e.g., Zheng
et al. (2015)). Nevertheless, we prove that these restrictions can be
completely removed by normalizing the LSS. More importantly, even if no limit
exists on the variance of the LSS, the new CLT could still hold.

  3. 3.

The entire technical part of this paper is built on the decomposition of the
LSS
$\sum_{j=1}^{p}f\left(\lambda_{j}\right)=\sum_{j=1}^{M}f\left(\lambda_{j}\right)+\sum_{j=M+1}^{p}f\left(\lambda_{j}\right)$.
Because the classical delta method cannot be applied to the unbounded part
$\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ and the bounded part
$\sum_{j=M+1}^{p}f\left(\lambda_{j}\right)$ is not a strict LSS of a sample
covariance matrix, the results of Bai and Silverstein (2004) and Jiang and Bai
(2021) cannot be adopted directly. In this paper, we leverage a ‘generalized
delta method’ and employ skillful transformations to prove the CLTs for the
unbounded and bounded parts, respectively. Moreover, we prove that the
unbounded and bounded parts are asymptotically independent, which leads to the
establishment of the new CLT.

  4. 4.

We verify that Roy’s largest root test is most powerful among the common tests
when the alternative (1.6) has only one spiked eigenvalue, which has also been
mentioned by Olson (1974); Johnstone and Nadler (2017). Furthermore, we
demonstrate that when the number of spikes is larger than one, the LSSs could
exhibit higher asymptotic power than Roy’s largest root test when the
divergence rates of spiked eigenvalues are higher than $\sqrt{n}$.

The remaining sections are organized as follows: Section 2 presents a detailed
description of our notations and assumptions. The main results for the CLT for
the LSSs of sample covariance matrices are stated in Section 3. In Section 4,
we explore an application of our main results. We also report the results of
numerical studies in Section 5. Technical proofs are presented in Section 6.
This paper is also accompanied by an online supplementary file that includes
the following materials: (i) some postponed proofs for Theorems 3.1–4.5; (ii)
some additional simulation results, and (iii) some useful lemmas.

##  2 Notations and assumptions

Throughout the paper, we use bold capital letters and bold italic lowercase
letters to represent matrices and vectors, respectively. Scalars are
represented by regular letters. $\bm{e}_{i}$ denotes a standard basis vector
whose components are all zero, except the $i$-th component, which is equal to
1. We use tr$({\bf A})$, ${\bf A}^{\top}$ and ${\bf A}^{\ast}$ to denote the
trace, transpose and conjugate transpose of matrix ${\bf A}$, respectively. We
also use $f^{\prime}$ to denote the derivative of function $f$, and we use
$\frac{\partial}{\partial z_{1}}f(z_{1},z_{2})$ to denote the partial
derivative of function $f$ with respect to $z_{1}$. Let $\left[{\bf
A}\right]_{ij}$ denote the $(i,j)$-th entry of the matrix ${\bf A}$ and
$\oint_{\mathcal{C}}f(z)dz$ denote the contour integral of $f(z)$ on the
contour $\mathcal{C}$. Let $\lambda_{i}^{{\bf A}}$ be the $i$th largest
eigenvalue of matrix ${\bf A}$. Weak convergence is denoted by
$\stackrel{{\scriptstyle d}}{{\rightarrow}}$. Throughout this paper, we use
$o(1)$ (resp. $o_{p}(1)$) to denote a negligible scalar (resp. in
probability), and the notation $C$ represents a generic constant that may vary
from line to line.

We adopt the notation ${\bf X}=(\bm{x}_{1},\ldots,\bm{x}_{n})=(x_{ij})$,
$1\leq i\leq p$, $1\leq j\leq n$. Let $\rho_{1}\geq\cdots\geq\rho_{p}$ be the
eigenvalues of $\boldsymbol{\Sigma}$ and the singular value decomposition of
${\bf T}$ be

${\bf T}={\bf V}{\bf D}^{1/2}{\bf U}^{*}=({\bf V}_{1},{\bf
V}_{2})\left(\begin{array}[]{cc}{\bf D}_{1}^{\frac{1}{2}}&0\\\ 0&{\bf
D}_{2}^{\frac{1}{2}}\end{array}\right)({\bf U}_{1},{\bf U}_{2})^{\ast}.$ (2.7)

Here ${\bf U}$ and ${\bf V}$ are unitary matrices, and ${\bf
D}_{1}={\textrm{diag}}(\underbrace{\alpha_{1},\dots,\alpha_{1}}_{d_{1}},\underbrace{\alpha_{2},\dots,\alpha_{2}}_{d_{2}},\dots,\underbrace{\alpha_{K},\dots,\alpha_{K}}_{d_{K}})$
is a diagonal matrix whose diagonal elements tend to infinity. To avoid
confusion, we refer to $\\{\alpha_{i},i=1,\dots,K\\}$ as the diverging spikes
in the following.  Assume $d_{1}+\cdots+d_{K}=M$. ${\bf D}_{2}$ is the
diagonal matrix of the eigenvalues with bounded components, including bounded
spiked eigenvalues and bulk eigenvalues. Moreover, let $d_{0}=0$ and
$J_{k}=\left\\{\sum_{i=0}^{k-1}d_{i}+1,\ldots,\sum_{i=0}^{k}d_{i}\right\\}$,
thus $\rho_{i}=\alpha_{k}$ if $i\in J_{k}$. Then, the corresponding sample
covariance matrix ${\bf B}=\frac{1}{n}{\bf T}{\bf X}{\bf X}^{\ast}{\bf
T}^{\ast}$ is the so-called generalized spiked sample covariance matrix.
Corresponding to the decomposition of ${\bf D}$, we decompose ${\bf
V}=\left({\bf V}_{1},{\bf V}_{2}\right)$, ${\bf U}=\left({\bf U}_{1},{\bf
U}_{2}\right)$, and denote ${\bf\Gamma}={\bf V}_{2}{\bf D}_{2}^{1/2}{\bf
U}_{2}^{\ast}$, $\bm{r}_{j}=\frac{1}{\sqrt{n}}{\bf\Gamma}\bm{x}_{j}$, and
${\bf A}_{j}=\frac{1}{n}{\bf\Gamma}{\bf X}{\bf X}^{*}{\bf\Gamma}^{*}-z{\bf
I}-\bm{r}_{j}\bm{r}_{j}^{*}.$ Let $\mathbb{E}_{j}$ be the conditional
expectation with respect to the $\sigma$-field generated by
$\bm{r}_{1},\dots,\bm{r}_{j}$. For any matrix ${\bf A}$ with real eigenvalues,
the empirical spectral distribution of ${\bf A}$ is denoted by

$F^{{\bf A}}\left(x\right)=\frac{1}{p}\left(\text{number of eigenvalues of
}{\bf A}\leq x\right).$

For any function of bounded variation $F$ on the real line, its Stieltjes
transform is defined by

$m_{F}(z)=\int\frac{1}{\lambda-z}\mathrm{~{}d}F(\lambda),\quad
z\in\mathbb{C}^{+}:=\\{z\in\mathbb{C}:\Im z>0\\}.$

The assumptions used to obtain the results in this paper are as follows:

######  Assumption 1.

$\\{x_{ij},1\leq i\leq p,1\leq j\leq n\\}$ are i.i.d. random variables with
common moments

$\mathbb{E}x_{ij}=0,\quad\mathbb{E}\left|x_{ij}\right|^{2}=1,\quad\beta_{x}=\mathbb{E}\left|x_{ij}\right|^{4}-\left|\mathbb{E}x_{ij}^{2}\right|^{2}-2,\quad\alpha_{x}=\left|\mathbb{E}x_{ij}^{2}\right|^{2}.$

######  Assumption 2.

As $\min\\{p,n\\}\to\infty$, the ratio of the dimension-to-sample size (RDS)
$c_{n}:={p}/{n}\rightarrow c>0.$

######  Remark 2.1.

Assumptions 1 and 2 are standard in RMT. If $\mathbb{E}x_{ij}\neq 0$, we can
use the centralized sample covariance matrices and $n-1$ instead of ${\bf
B}_{n}$ and $n$, respectively, and the following results also hold. Details
can be found in (Zheng et al., 2015). Therefore, in the sequel, we assume that
$\mathbb{E}x_{ij}=0$ without loss of generality.

######  Assumption 3.

${\bf T}$ is nonrandom. As $\min\\{p,n\\}\to\infty$, $\alpha_{K}\to\infty$ and
$H_{n}:=F^{{\bf\Gamma}{\bf\Gamma}^{*}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}H$, where $H$ is a distribution function on the real line.
$M$ is fixed.

######  Remark 2.2.

Similar to Silverstein (1995) that under Assumptions 1-3, $F^{{\bf
B}}\stackrel{{\scriptstyle d}}{{\rightarrow}}F^{c,H}$ almost surely, where
$F^{c,H}$ is a nonrandom distribution function whose Stieltjes transform
$m:=m_{F^{c,H}}(z)$ satisfies the following equation:

$\displaystyle m=\int\frac{1}{t(1-c-czm)-z}dH(t).$ (2.8)

In the sequel, we call $F^{c,H}$ the limiting spectral distribution (LSD) of
${\bf B}$. Moreover, because the matrix $\underline{{\bf B}}=\frac{1}{n}{\bf
X}^{*}{\bf T}^{*}{\bf T}{\bf X}$ shares the same nonzero eigenvalues as ${\bf
B}$, equation (2.8) can be rewritten as

$\underline{m}=-\left(z-c\int\frac{t}{1+t\underline{m}}dH(t)\right)^{-1},$

where $\underline{m}:=m_{\underline{F}^{c,H}}(z)$ represents the Stieltjes
transform of the LSD of $\underline{{\bf B}}$.

######  Assumption 4.

Test functions $f_{1},\dots,f_{h}$ are analytic on a connected open region of
the complex plane containing the support of $F^{c_{n},H_{n}}$ for almost all
$n.$ Moreover, we suppose that for any $l=1,\dots,h$,

$\lim_{\\{x_{n},y_{n}\\}\to\infty\atop{x_{n}}/{y_{n}}\rightarrow
1}\frac{f_{l}^{\prime}\left(x_{n}\right)}{f_{l}^{\prime}\left(y_{n}\right)}=1.$

######  Remark 2.3.

In fact, Assumption 4 is not highly restrictive in practice, as many common
functions such as logarithmic and polynomial functions satisfy it. However, it
is worth noting that the exponential function does not satisfy this
assumption.

For convenience of description, we introduce some notations before presenting
the main results in the next section. Let $\underline{F}^{c,H}$ denote the LSD
of matrix $n^{-1}{\bf X}^{\ast}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf X}$,
${\bf U}_{1}=\left(u_{ij}\right)_{i=1,\dots,p;j=1,\dots,M}$,
$\mathcal{U}_{i_{1}j_{1}i_{2}j_{2}}=\sum_{t=1}^{p}\overline{u}_{ti_{1}}u_{tj_{1}}u_{ti_{2}}\overline{u}_{tj_{2}}$,
$\phi_{n}\left(x\right)=x\left(1+c_{n}\int\frac{t}{x-t}dH_{n}\left(t\right)\right)$,

$\phi_{k}=\phi\left(x\right)\mid_{x=\alpha_{k}}=\alpha_{k}\left(1+c\int\frac{t}{\alpha_{k}-t}dH\left(t\right)\right),~{}~{}\theta_{k}=\phi_{k}^{2}\underline{m}_{2}\left(\phi_{k}\right),~{}~{}\nu_{k}=\phi_{k}^{2}\underline{m}^{2}\left(\phi_{k}\right),$
$\underline{m}\left(\lambda\right)=\int\frac{1}{x-\lambda}d\underline{F}^{c,H}\left(x\right),~{}~{}\underline{m}_{2}\left(\lambda\right)=\int\frac{1}{\left(\lambda-x\right)^{2}}d\underline{F}^{c,H}\left(x\right),$
$c_{nM}=\dfrac{p-M}{n},~{}~{}H_{2n}=F^{{\bf D}_{2}},~{}~{}{\bf
P}_{n}(z)=\left((1-c_{nM}){\bf\Gamma}{\bf\Gamma}^{*}-zc_{nM}m_{2n0}(z){\bf\Gamma}{\bf\Gamma}^{*}-z{\bf
I}_{p}\right)^{-1},$
$\varpi_{nkl}=\frac{\phi_{n}\left(\alpha_{k}\right)}{\sqrt{n}}f_{l}^{\prime}\left(\phi_{n}\left(\alpha_{k}\right)\right),~{}~{}s_{k}^{2}=\frac{\left(\alpha_{x}+1\right)d_{k}}{\theta_{k}}+\frac{\beta_{x}\nu_{k}\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}{\theta_{k}^{2}},$
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}),$
$\Theta_{0,n}(z_{1},z_{2})=\dfrac{\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{(\underline{m}_{2n0}(z_{1})-\underline{m}_{2n0}(z_{2}))^{2}}-\dfrac{1}{(z_{1}-z_{2})^{2}},$
$\Theta_{1,n}(z_{1},z_{2})=\frac{\partial}{\partial
z_{2}}\left\\{\dfrac{\partial\mathcal{A}_{n}(z_{1},z_{2})}{\partial
z_{1}}\dfrac{1}{1-\alpha_{x}\mathcal{A}_{n}(z_{1},z_{2})}\right\\},$
${\mathcal{A}_{n}(z_{1},z_{2})}=\dfrac{z_{1}z_{2}}{n}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\mathrm{tr}{{\bf\Gamma}^{*}{\bf
P}_{n}(z_{1}){\bf\Gamma}{\bf\Gamma}^{\top}{\bf
P}_{n}(z_{2})^{\top}\bar{{\bf\Gamma}}},$
$\Theta_{2,n}(z_{1},z_{2})=\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii},$ $\displaystyle\mu_{l}=$
$\displaystyle-\frac{\alpha_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}\frac{c_{nM}f_{l}(z)\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{\left(1-c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)\left(1-\alpha_{x}c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)}dz$
$\displaystyle-\frac{\beta_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}\frac{c_{nM}f_{l}(z)\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{1-c_{nM}\int\underline{m}^{2}_{2n0}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-2}dH_{2n}(t)}dz,\quad
l=1,\dots,h.$

Here, $m_{2n0}(z)$ is the Stieltjes transform of $F^{c_{nM},H_{2n}}$, where
$F^{c_{nM},H_{2n}}$ is the LSD $F^{c,H}$ with $\\{c,~{}H\\}$ replaced by
$\\{c_{nM},~{}H_{2n}\\}$,
$\underline{m}_{2n0}(z)=-\frac{1-c_{nM}}{z}+c_{nM}m_{2n0}(z)$ and
$\mathcal{C}$ is a closed contour in the complex plane enclosing the support
of $F^{c_{n},H_{n}}$ and it is also enclosed in the analytic area of $f_{l}$.
For clarity, $m_{1n0}(z)$ denotes the Stieltjes transform of
$F^{c_{n},H_{n}}$, $m_{n}=\frac{1}{p}\mathrm{tr}\left({\bf B}-z{\bf
I}_{p}\right)^{-1}$, and $m_{2n}=\frac{1}{p-M}\mathrm{tr}\left({\bf
S}_{22}-z{\bf I}_{p-M}\right)^{-1}$.

Note that

$\sum_{j=1}^{p}f\left({\bf\lambda}_{j}\right)=p\int f\left(x\right)dF^{{\bf
B}}(x).$

Thus, for brevity, we define the normalized LSSs as

$Y_{l}=\int
f_{l}\left(x\right)dG_{n}\left(x\right)-{\sum_{k=1}^{K}d_{k}f_{l}\left(\phi_{n}\left(\alpha_{k}\right)\right)}-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{l}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz,\quad
l=1,2,\dots,h,$

where

$G_{n}\left(x\right)=p[F^{{\bf
B}}\left(x\right)-F^{c_{n},H_{n}}\left(x\right)].$

##  3 Main results

Now, we are in a position to present our main theorems and their proofs are
provided in Section 6 and the supplementary material. We first establish a CLT
for an LSS without any restrictions imposed on the Gaussian moments or on the
structures of the population covariance matrix by normalizing the LSS.

######  Theorem 3.1.

Under Assumptions 1–4, we have

$\frac{Y_{1}-\mu_{1}}{\varsigma_{1}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,1\right),$

where

$\displaystyle\varsigma_{1}^{2}=\sum_{k=1}^{K}\varpi_{nk1}^{2}s_{k}^{2}-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{1}\left(z_{1}\right)f_{1}\left(z_{2}\right)\vartheta_{n}^{2}dz_{1}dz_{2},$
(3.9)

$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are nonoverlapping and closed contours
in the complex plane enclosing the support of $F^{c_{n},H_{n}}$.
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are also enclosed in the analytic area
of $f_{1}.$

######  Remark 3.1.

Recall the definitions
$s_{k}^{2}=\frac{\left(\alpha_{x}+1\right)d_{k}}{\theta_{k}}+\frac{\beta_{x}\nu_{k}\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}{\theta_{k}^{2}}$ and
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}).$
Notably, the term $\Theta_{0,n}(z_{1},z_{2})$ has a limitation under
Assumptions 1–4, which has already been discussed in Bai and Silverstein
(2004). Additionally, if $\boldsymbol{\Sigma}$ is complex, the convergence of
$\Theta_{1,n}(z_{1},z_{2})$ is not guaranteed. The term
$\Theta_{2,n}(z_{1},z_{2})$ involves the quantities $\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{i}){\bf\Gamma}\right]_{ii}$, which depend not only on the
eigenvalues of ${\bf D}_{2}$ but also on their associated eigenvectors.
Furthermore, the term $s_{k}^{2}$ indicates that the variance is influenced by
the second and fourth moments of $x_{ij}$, spiked eigenvalues, and their
associated eigenvectors. The limit of $\varpi_{nk1}^{2}$ is allowed to be
infinite.

######  Remark 3.2.

After a closer look at the variance (3.9), we can also find that the first
part of the formula (3.9) is the variance containing diverging spikes and the
second part is the variance containing the bounded eigenvalues. When
$\phi_{n}\left(\alpha_{1}\right)f^{\prime}_{1}\left(\phi_{n}\left(\alpha_{1}\right)\right)=o\left(\sqrt{n}\right),$
the first term in formula (3.9) tends to 0, and the variance is mainly
affected by the second part. When
$\phi_{n}\left(\alpha_{1}\right)f^{\prime}_{1}\left(\phi_{n}\left(\alpha_{1}\right)\right)$
is of order $\sqrt{n}$, the two parts of formula (3.9) are of the same order,
and the variance is affected by both. When the order of
$\phi_{n}\left(\alpha_{1}\right)f^{\prime}_{1}\left(\phi_{n}\left(\alpha_{1}\right)\right)$
is higher than $\sqrt{n}$, the first part of formula (3.9) is much larger than
the second part; therefore, the spiked part dominates the variance value.

As a minor price for the removal of the bounded spectrum condition, the new
CLT described above applies only to a single LSS. To guarantee that the new
CLT will apply to multiple normalized LSSs, structural assumptions about the
population covariance matrices are needed.

######  Assumption 5.

${\bf T}$ is real or the variables $x_{ij}$ are complex satisfying
$\alpha_{x}=0.$

######  Assumption 6.

${\bf T}^{*}{\bf T}$ is diagonal or $\beta_{x}=0.$

######  Remark 3.3.

Assumptions 5 and 6 are used as a replacement for the Gaussian-like moments
condition. It has been proved by Zheng et al. (2015) that these two structural
assumptions regarding the population matrices are necessary for their results
when the Gaussian-like moments condition in the BST does not hold.

The following theorem is a nontrivial extension of the BST:

######  Theorem 3.2.

Under Assumptions 1–6, the random vector

$\left(\frac{Y_{1}-\mu_{1}}{\sigma_{1}},\dots,\frac{Y_{h}-\mu_{h}}{\sigma_{h}}\right)^{\top}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{h}\left(0,{\bf\Psi}\right),$

with variance

$\displaystyle\sigma_{l}^{2}$
$\displaystyle=\sum_{k=1}^{K}\varpi_{nkl}^{2}s_{k}^{2}-\kappa_{nll},\quad
l=1,\dots,h,$

and covariance matrix ${\bf\Psi}=\left(\psi_{st}\right)_{h\times h}$, where
$\psi_{st}=\lim_{n\to\infty}\psi_{nst}$,

$\displaystyle\psi_{nst}=\frac{\sum_{k=1}^{K}\varpi_{nks}\varpi_{nkt}s_{k}^{2}-\kappa_{nst}}{\sqrt{\sum_{k=1}^{K}\varpi_{nks}^{2}s_{k}^{2}-\kappa_{nss}}\sqrt{\sum_{k=1}^{K}\varpi_{nkt}^{2}s_{k}^{2}-\kappa_{ntt}}},$
$\displaystyle\kappa_{nst}$
$\displaystyle=\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\frac{f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)}{\left(\underline{m}_{2n0}\left(z_{1}\right)-\underline{m}_{2n0}\left(z_{2}\right)\right)^{2}}d\underline{m}_{2n0}\left(z_{1}\right)d\underline{m}_{2n0}\left(z_{2}\right)$
$\displaystyle+\frac{c_{nM}\beta_{x}}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\int\frac{f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)t^{2}}{\left(\underline{m}_{2n0}\left(z_{1}\right)t+1\right)^{2}\left(\underline{m}_{2n0}\left(z_{2}\right)t+1\right)^{2}}dH_{2n}(t)d\underline{m}_{2n0}\left(z_{1}\right)d\underline{m}_{2n0}\left(z_{2}\right)$
$\displaystyle\quad+\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)\left[\frac{\partial^{2}}{\partial
z_{1}\partial
z_{2}}\log\left(1-a_{n}\left(z_{1},z_{2}\right)\right)\right]dz_{1}dz_{2},$

and

$\displaystyle
a_{n}\left(z_{1},z_{2}\right)=\alpha_{x}\left(1+\frac{\underline{m}_{2n0}\left(z_{1}\right)\underline{m}_{2n0}\left(z_{2}\right)\left(z_{1}-z_{2}\right)}{\underline{m}_{2n0}\left(z_{2}\right)-\underline{m}_{2n0}\left(z_{1}\right)}\right).$

######  Remark 3.4.

Define
$\tilde{\psi}_{nst}=\dfrac{\sum_{k=1}^{K}\varpi_{nks}\varpi_{nkt}s_{k}^{2}-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)\vartheta_{n}^{2}dz_{1}dz_{2}}{\varsigma_{s}\varsigma_{t}}.$
If $\bm{\tilde{\Psi}}_{n}=(\tilde{\psi}_{nst})_{h\times h}$ is invertible for
all sufficiently large $n$, we conjecture that, similar to Theorem 3.1, the
convergence

$\bm{\tilde{\Psi}}_{n}^{-1/2}\left(\frac{Y_{1}-\mu_{1}}{\sigma_{1}},\dots,\frac{Y_{h}-\mu_{h}}{\sigma_{h}}\right)^{\top}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{h}\left(0,{\bf I}_{h}\right)$

holds without requiring Assumptions 5 and 6. It should be noted that
$\bm{\tilde{\Psi}}_{n}$ is singular if the set of test functions is linearly
dependent. However, determining the invertibility of $\bm{\tilde{\Psi}}_{n}$
becomes challenging when the test functions are completely linearly
independent. Hence, the extension to the removal of Assumptions 5 and 6 in
Theorem 3.2 is left for future work.

######  Remark 3.5.

If $\varpi_{nkl}\rightarrow 0$ as $n\rightarrow\infty$, Theorem 3.2 coincides
with Theorem 2.1 of Zheng et al. (2015). If the test functions $f_{l}=x$ and
$x^{2}$, then Theorem 3.2 reduces to Theorem 2.1 of Yin (2021). Notably, the
results in Yin (2021) required higher-order moment conditions.

##  4 Application

In this section, we focus on a hypothesis test concerning whether the
population covariance matrix $\boldsymbol{\Sigma}$ is equal to the identity
matrix or a spiked model, i.e.,

$\displaystyle H_{0}:\boldsymbol{\Sigma}={\bf I}_{p}\quad\text{vs.}\quad
H_{1}:\boldsymbol{\Sigma}=\mathbf{V}\left(\begin{array}[]{cc}{\bf D}_{1}&0\\\
0&{\bf I}_{p-M}\end{array}\right)\mathbf{V}^{\ast},$ (4.12)

where ${\bf D}_{1}$ is a diagonal matrix of the diverging spiked eigenvalues
of $\boldsymbol{\Sigma}$. There are several classical test statistics for this
problem, but due to the limited length of this paper, we only consider the
likelihood ratio (LR) test statistic (Wilks, 1938) and the Nagao’s trace (NT)
test statistic (Nagao, 1973) in this section. Specifically, the LR and NT
statistics can be formulated as

$L=\operatorname{tr}{\bf B}-\log\left|{\bf
B}\right|-p~{}~{}\mbox{and}~{}~{}W=\mathrm{tr}({\bf B}-{\bf I}_{p})^{2},$

respectively. Under the null hypothesis, the asymptotic properties of the LR
and NT statistics for high-dimensional settings have been investigated
extensively in the literature; here, we refer to Bai et al. (2009); Jiang and
Yang (2013); Ledoit and Wolf (2002); Wang and Yao (2013); Onatski et al.
(2013) for more details. Thus, in this section, we mainly focus on the
alternative hypothesis. However, to provide better comparisons, we also
present the asymptotic distributions under the null hypothesis in the
following theorems.

###  4.1 Asymptotic results for the LR and NT statistics

In this subsection, we present the asymptotic results for LR and NT test
statistics for the testing problem (4.12).

######  Theorem 4.1 (CLT for the LR statistic).

Under Assumptions 1–4 with $c_{n}=p/n\rightarrow c\in(0,1)$, we have

  * •

(Under $H_{0}$)

$\dfrac{L-p\ell_{l}-\mu_{l}}{\varsigma_{l}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

where

$\displaystyle\ell_{l}=1-\frac{c_{n}-1}{c_{n}}\log\left(1-c_{n}\right),~{}~{}~{}\mu_{l}=-\frac{\log\left(1-c_{n}\right)}{2}\alpha_{x}+\frac{c_{n}}{2}\beta_{x}$

and

$\varsigma_{l}^{2}=(\alpha_{x}+1)(-\log\left(1-c_{n}\right)-c_{n}).$

  * •

(Under $H_{1}$)

$\dfrac{L-(p-M)\breve{\ell}_{l}-\breve{\mu}_{l}}{\breve{\varsigma}_{l}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

where

$\displaystyle\breve{\ell}_{l}=$ $\displaystyle
1-\frac{c_{nM}-1}{c_{nM}}\log\left(1-c_{nM}\right),~{}\breve{\mu}_{l}=-\frac{\log\left(1-c_{nM}\right)}{2}\alpha_{x}+\frac{c_{nM}}{2}\beta_{x}$
$\displaystyle+\sum_{k=1}^{K}d_{k}\left(\phi_{n}\left({\alpha}_{k}\right)-\log\phi_{n}\left({\alpha}_{k}\right)-1\right)-M(c_{nM}+\log(1-c_{nM}))$

and

$\displaystyle\breve{\varsigma}_{l}^{2}=$
$\displaystyle\sum_{k=1}^{K}\frac{\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(-\log(1-c_{nM})-c_{nM}\right).$

######  Remark 4.1.

If $c\geq 1$, then ${\bf B}_{n}$ could be singular for large $n$, which would
give rise to an undefined LR statistic $L$. Thus, the additional restriction
$c<1$ is added in Theorem 4.1.

######  Remark 4.2.

Note that $\phi_{n}\left({\alpha}_{k}\right)$ and $s_{k}$ are defined in
Section 2. Under the alternative hypothesis $H_{1}$ in (4.12), we can adopt
simplification $\phi_{n}\left(\alpha_{k}\right)=\alpha_{k}+c_{nM}+o(1)$, and

$s_{k}^{2}={\left(\alpha_{x}+1\right)d_{k}}+{\beta_{x}\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}+o(1).$

######  Theorem 4.2 (CLT for the NT statistic).

Under Assumptions 1–4, we have

  * •

(Under $H_{0}$)

$\frac{W-pc_{n}-\mu_{w}}{\varsigma_{w}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

where

$\displaystyle\mu_{w}=c_{n}(\alpha_{x}+\beta_{x})~{}~{}\mbox{and}~{}~{}\varsigma_{w}^{2}=(\alpha_{x}+1)(4c_{n}^{3}+2c_{n}^{2})+4\beta_{x}c_{n}^{3}.$

  * •

(Under $H_{1}$)

$\displaystyle\frac{W-(p-M)c_{nM}-\breve{\mu}_{w}}{\breve{\varsigma}_{w}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

where

$\displaystyle\breve{\mu}_{w}=$ $\displaystyle
c_{nM}(\alpha_{x}+\beta_{x})+\sum_{k=1}^{K}d_{k}\left(\phi_{n}(\alpha_{k})-1\right)^{2}-Mc_{nM}^{2}$

and

$\displaystyle\breve{\varsigma}_{w}^{2}=$
$\displaystyle\sum_{k=1}^{K}\frac{4\phi_{n}^{2}\left(\alpha_{k}\right)\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(4c_{nM}^{3}+2c_{nM}^{2}\right)+4\beta_{x}c_{nM}^{3}.$

######  Remark 4.3.

From the covariance terms $\breve{\varsigma}_{l}^{2}$ and
$\breve{\varsigma}_{w}^{2}$, one can find that these CLTs are related to the
components of the right singular vectors $\mathbf{U}$, but not to the left
singular vectors $\mathbf{V}$. Furthermore, since ${\bf D}_{2}$ is an identity
matrix, ${\bf U}_{2}$ does not affect the asymptotic CLTs. Therefore, the only
singular vectors of ${\bf T}$ affecting the results are ${\bf U}_{1}$, which
are involved in $s_{k}^{2}$.

The proofs of Theorems 4.1 and 4.2 are given in the supplementary material. To
avoid confusion with the classical distributions of the LR test and NT test,
we refer to the CLTs above as the corrected LR test (CLRT) and corrected NT
test (CNTT) in the sequel. From Theorems 4.1 and 4.2, we reject the null
hypothesis $H_{0}$ in (4.12) if

$L>z_{\xi}{\varsigma_{l}}+p\ell_{l}+\mu_{l}$

and

$W>z_{\xi}{\varsigma_{w}}+c_{n}(p+\alpha_{x}+\beta_{x}),$

where $\xi$ is the significance level of the test and $z_{\xi}$ is the $1-\xi$
quantile of the standard Gaussian distribution $\Phi$. For the power functions
of CLRT and CNTT, we have the following theorems.

######  Theorem 4.3 (Power function of CLRT).

Under Assumptions 1–4 with $c_{n}=p/n\rightarrow c\in(0,1)$ and $H_{1}$ in
(4.12), we have that the power function of the CLRT
$P_{L}=P(L>z_{\xi}{\varsigma_{l}}+p\ell_{l}+\mu_{l})$ satisfies

$\displaystyle
P_{L}-\Phi\left(\frac{\sum_{k=1}^{K}d_{k}\left(\phi_{n}\left({\alpha}_{k}\right)-\log\phi_{n}\left({\alpha}_{k}\right)\right)-M(1+c)-z_{\xi}{\varsigma_{l}}}{\sqrt{\sum_{k=1}^{K}\frac{\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(-\log(1-c)-c\right)}}\right)\rightarrow
0,$ (4.13)

as $n\to\infty$.

######  Theorem 4.4 (Power function of CNTT).

Under Assumptions 1–4 and $H_{1}$ in (4.12), we have that the power function
of the CNTT $P_{W}=P(W>z_{\xi}{\varsigma_{w}}+c_{n}(p+\alpha_{x}+\beta_{x}))$
satisfies

$\displaystyle
P_{W}-\Phi\left(\frac{\sum_{k=1}^{K}d_{k}\left(\phi_{n}(\alpha_{k})-1\right)^{2}-Mc^{2}-2Mc-
z_{\xi}{\varsigma_{w}}}{\sqrt{\sum_{k=1}^{K}\frac{4\phi_{n}^{2}\left(\alpha_{k}\right)\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(4c^{3}+2c^{2}\right)+4\beta_{x}c^{3}}}\right)\rightarrow
0,$ (4.14)

as $n\to\infty$.

######  Remark 4.4.

Since $s_{k}^{2}$ is nonrandom and of order $O(1)$, $P_{L}$ and $P_{W}$ tend
to 1 as $\alpha_{1}\to\infty$. The detailed analysis of the power functions of
$P_{L}$ and $P_{W}$ is discussed in the next subsection.

###  4.2 Power analysis

This subsection discusses the power functions of $P_{L}$ and $P_{W}$. For
simplicity, in this subsection, we assume that $\\{x_{ij}\\}$ are real, i.e.,
$\alpha_{x}=1$. We first derive the asymptotic power of Roy’s largest root
test (RLRT) to detect $H_{1}$ in (4.12) for comparison. Recall RLRT statistic
$\lambda_{1}.$ Under Assumptions 1–4 and $H_{0}$ in (4.12), it follows from
Theorem 2.7 of Ding and Yang (2018) that

$\displaystyle\frac{\lambda_{1}-\mu_{r}}{\varsigma_{r}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}F_{TW},$

where $\mu_{r}=\left(1+\sqrt{c_{n}}\right)^{2},$
$\varsigma_{r}=n^{-2/3}\left(1+\sqrt{c_{n}}\right)\left(1+\sqrt{c_{n}^{-1}}\right)^{1/3}$
and $F_{TW}$ is the Type 1 Tracy-Widom (TW) distribution. Let $t_{\xi}$ be the
$1-\xi$ quantile of TW distribution with significance level $\xi$. Then we
have the following theorem regarding the power function of RLRT.

######  Theorem 4.5 (Power function of RLRT).

Under Assumptions 1–4 and $H_{1}$ in (4.12), if the multiplicity of
$\alpha_{1}$ is one, then the power function of the RLRT
$P_{R}=\mathbb{P}(\lambda_{1}>t_{\xi}{\varsigma_{r}}+\mu_{r})$ satisfies

$\displaystyle
P_{R}-\Phi\left(-\frac{t_{\xi}{\varsigma_{r}}+{\mu_{r}}-\phi_{n}\left(\alpha_{1}\right)}{s_{1}\phi_{n}\left(\alpha_{1}\right)/\sqrt{n}}\right)\rightarrow
0,$ (4.15)

as $n\to\infty$.

The proof of this theorem is postponed to the supplementary material. It is
clear that if $\alpha_{1}>1+\sqrt{c}$ uniformly, then $P_{R}\to 1$ as
$n\to\infty$. According to Anderson (2003), compared with the classical LSSs,
RLRT has the highest asymptotic power to detect rank-one alternatives and
under low dimensional settings. This property has also been demonstrated by
Olson (1974) and Johnstone and Nadler (2017). In the following, we discuss the
asymptotic power functions of CLRT, CNTT and RLRT. In particular, we will show
that except for the case in which the number of spikes is equal to 1, CLRT and
CNTT may exhibit higher asymptotic power than RLRT in some scenarios.

Define

$\displaystyle\begin{split}\varkappa_{L}&=\frac{\sum_{k=1}^{K}d_{k}\left(\phi_{n}\left({\alpha}_{k}\right)-\log\phi_{n}\left({\alpha}_{k}\right)\right)-M(1+c)-z_{\xi}{\varsigma_{l}}}{\sqrt{\sum_{k=1}^{K}\frac{\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}-2\left(\log(1-c)+c\right)}}\\\
\varkappa_{W}&=\frac{\sum_{k=1}^{K}d_{k}\left(\phi_{n}(\alpha_{k})-1\right)^{2}-Mc^{2}-2Mc-
z_{\xi}{\varsigma_{w}}}{\sqrt{\sum_{k=1}^{K}\frac{4\phi_{n}^{2}\left(\alpha_{k}\right)\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+2\left(4c^{3}+2c^{2}\right)+4\beta_{x}c^{3}}}\\\
\varkappa_{R}&=\frac{\phi_{n}\left(\alpha_{1}\right)-{\mu_{r}}-t_{\xi}{\varsigma_{r}}}{s_{1}\phi_{n}\left(\alpha_{1}\right)/\sqrt{n}}.\end{split}$
(4.16)

Since CLRT, CNTT and RLRT statistics are all asymptotically normally
distributed under the alternative hypothesis, according to formulas
(4.13)–(4.15), comparing the convergence rates of power functions $P_{L}$,
$P_{W}$, and $P_{R}$ is equivalent to comparing the divergence rates
$\varkappa_{L}$, $\varkappa_{W}$ and $\varkappa_{R}$ tend to infinity. Note
that $\\{z_{\xi}{\varsigma_{l}}$, $z_{\xi}{\varsigma_{w}}$,
$t_{\xi}{\varsigma_{r}}$} are all of order $O(1)$, $\\{K,M\\}$ are fixed,
$0<c<1$, $\phi_{n}(\alpha_{k})=\alpha_{k}+c+o(1)$ and
$s_{k}^{2}={2d_{k}}+{\beta_{x}\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}+o(1)$. In the sequel, we use the
notations $A_{n}=\Omega(B_{n})$, $A_{n}\simeq B_{n}$ and $A_{n}\asymp B_{n}$
to denote $B_{n}=O(A_{n})$, $A_{n}=B_{n}+o(B_{n})$ and
$C^{-1}A_{n}<B_{n}<CA_{n}$, respectively, for some constant $C>1$. Then, we
have the following conclusions.

  * •

($M=1$) For $M=1$, i.e., there is only one diverging spike, we report the
divergence rates of $\varkappa_{L}$, $\varkappa_{W}$ and $\varkappa_{R}$ in
Table 1. From these results, we can conclude that the RLRT is asymptotically
more powerful than CLRT and CNTT whenever $\alpha_{1}\to\infty$. Here, one
should note that $1+2c+\beta_{x}c>0$ and $\log(1-c)+c<0$ provided that $c<1$.
Moreover, if $\alpha_{1}=o(n^{1/2})$, then the divergence rate of
$\varkappa_{W}$ is higher than $\varkappa_{L}$. However, when
$\alpha_{1}=\Omega(n^{1/2})$, $\varkappa_{L}$ could be larger than
$\varkappa_{W}$, such as $n=o(\alpha_{1}^{2})$.

Table 1: Divergence rates of $\varkappa_{L}$, $\varkappa_{W}$ and $\varkappa_{R}$ when $M=1$ | $\varkappa_{L}$ | $\varkappa_{W}$ | $\varkappa_{R}$
---|---|---|---
$\alpha_{1}=o(n^{\frac{1}{4}})$ | $\asymp\alpha_{1}$ | $\asymp\alpha_{1}^{2}$ | $\simeq\frac{\sqrt{n}}{s_{1}}$
$\Omega(n^{\frac{1}{4}})=\alpha_{1}=o(n^{\frac{1}{2}})$ | $\asymp\alpha_{1}$ | $\simeq\frac{\sqrt{n}}{2\sqrt{s_{1}^{2}+c^{2}(1+2c+\beta_{x}c)n/\alpha_{1}^{4}}}$ | $\simeq\frac{\sqrt{n}}{s_{1}}$
$\alpha_{1}=\Omega(n^{\frac{1}{2}})$ | $\simeq\frac{\sqrt{n}}{\sqrt{s_{1}^{2}-2(\log(1-c)+c)n/\alpha_{1}^{2}}}$ | $\simeq\frac{\sqrt{n}}{2s_{1}}$ | $\simeq\frac{\sqrt{n}}{s_{1}}$
  * •

($M=2$) For $M=2$, we assume that the two diverging spikes are not equal,
i.e., $d_{1}=d_{2}=1$. In addition, for convenience of analysis, we assume
that the two spikes have the same divergence rate, i.e.,
$\alpha_{2}=k_{2}\alpha_{1}$ with some $k_{2}<1$. The results are presented in
Table 2. Since the power function of RLRT is relevant only to the largest
spikes and not to the other spikes, $\varkappa_{R}$ has the same result for
$M=2$ as it does for $M=1$, that is $\varkappa_{R}\simeq{\sqrt{n}}/{s_{1}}$.
Thus, we omit the results of $\varkappa_{R}$ in Table 2. We can conclude from
Table 2 that RLRT is asymptotically more powerful than CNTT whenever
$\alpha_{1}\to\infty$, because $1+k_{2}^{2}<2$ and $s_{2}\geq 0$. However, for
CLRT, if $n=o(\alpha_{1}^{2})$ and
$(s_{1}^{2}+k_{2}^{2}s_{2}^{2})/(1+k_{2})^{2}<s_{1}^{2}$, then $\varkappa_{L}$
could be larger than $\varkappa_{R}$. Since
$s_{k}^{2}=2+\beta_{x}\sum_{t=1}^{p}|u_{tk}|^{4}+o(1)$, with suitable values
of $\beta_{x}\geq-2$, $\sum_{t=1}^{p}|u_{tk}|^{4}\in[1/p,1]$ and $k_{2}<1$,
the inequality $(s_{1}^{2}+k_{2}^{2}s_{2}^{2})/(1+k_{2})^{2}<s_{1}^{2}$ can be
satisfied, such as choosing $\beta_{x}=0$. This property indicates that the
LSSs could exhibit higher asymptotic power than RLRT statistic in some special
scenarios.

Table 2: Divergence rates of $\varkappa_{L}$ and $\varkappa_{W}$ when $M=2$ and $\alpha_{2}=k_{2}\alpha_{1}$ | $\varkappa_{L}$ | $\varkappa_{W}$
---|---|---
$\alpha_{1}=o(n^{\frac{1}{4}})$ | $\asymp\alpha_{1}$ | $\asymp\alpha_{1}^{2}$
$\Omega(n^{\frac{1}{4}})=\alpha_{1}=o(n^{\frac{1}{2}})$ | $\asymp\alpha_{1}$ | $\simeq\frac{\sqrt{n}(1+k_{2}^{2})}{2\sqrt{s_{1}^{2}+k_{2}^{4}s_{2}^{2}+c^{2}(1+2c+\beta_{x}c)n/\alpha_{1}^{4}}}$
$\alpha_{1}=\Omega(n^{\frac{1}{2}})$ | $\simeq\frac{(1+k_{2})\sqrt{n}}{\sqrt{s_{1}^{2}+k_{2}^{2}s_{2}^{2}-2(\log(1-c)+c)n/\alpha_{1}^{2}}}$ | $\simeq\frac{\sqrt{n}(1+k_{2}^{2})}{2\sqrt{s_{1}^{2}+k_{2}^{4}s_{2}^{2}}}$
  * •

($M\geq 3$) For $M\geq 3$, the discussion is analogous to the cases of $M=1$
and $M=2$; thus, we omit the details because of space limitations. We merely
wish to emphasize here that when $M\geq 3$, CNTT is also potentially
asymptotically more powerful than RLRT. Suppose that
$\alpha_{t}=k_{t}\alpha_{1}$, $t=1,\dots,M$ and $1=k_{1}>k_{2}>\dots>k_{M}>0.$
It is not difficult to find that if $n=o(\alpha_{1}^{4})$, then
$\varkappa_{W}\simeq{\sqrt{n}\sum_{t=1}^{M}k_{t}^{2}}/{\sqrt{4\sum_{t=1}^{M}k_{t}^{4}s_{t}^{2}}}$,
which can be larger than $\varkappa_{R}\simeq{\sqrt{n}}/s_{1}$ with suitable
values of $k_{t}$ and $s_{t}$, $t=1,\dots,M$. For example, if $M=3$, we can
choose $k_{2}$ and $k_{3}$ close to 1, while choosing $s_{2}$ and $s_{3}$
close to 0. Notably, if $\beta_{x}=0$, e.g., $\\{x_{ij}\\}$ are Gaussian, then
$M$ must be at least 5 for $\varkappa_{W}>\varkappa_{R}$ to asymptotically
hold.

For illustration, we present some graphs of the functions $\varkappa_{L}$,
$\varkappa_{W}$ and $\varkappa_{R}$ in (4.16) with different numbers of
diverging spikes in Figure 1.

(a) $M=1$

(b) $M=2$

(c) $M=5$

Figure 1: Graphs of the functions $\varkappa_{L}$, $\varkappa_{W}$ and
$\varkappa_{R}$. We fix $(p,n,\xi,\beta_{x})=(100,300,0.05,0)$. The left panel
shows the curves for $M=1$. The middle panel shows the curves for $M=2$ with
$\alpha_{2}=0.9\alpha_{1}$. The right panel shows the curves for $M=5$ with
$\alpha_{2}=0.9\alpha_{1}$, $\alpha_{3}=0.85\alpha_{1}$,
$\alpha_{4}=0.8\alpha_{1}$ and $\alpha_{5}=0.75\alpha_{1}$.

##  5 Numerical studies

In this section, we report short numerical studies as an illustration of our
results. Our objective in the simulations is to examine the power analysis in
subsection 4.2.

We examine the following three different distributions of $x_{ij}:$

  * $Dt_{1}$:

$\left\\{x_{ij}\right\\}$ are i.i.d. samples from a standard Gaussian
population.

  * $Dt_{2}$:

$\left\\{x_{ij}\right\\}$ are i.i.d. samples from $Gamma(4,0.5)-2$.

  * $Dt_{3}$:

$\left\\{x_{ij}\right\\}$ are i.i.d. samples from Uniform population
distribution $U[-\sqrt{3},\sqrt{3}]$.

Note that in above settings, $\beta_{x}=0,\frac{3}{2},-\frac{6}{5},$
respectively.

In the current numerical studies, the null hypothesis is defined as
$H_{0}:\boldsymbol{\Sigma}={\bf I}_{p}.$ For the alternative hypothesis, we
adopt the following six population covariance matrix structures:

  * $H_{1}$:

$\boldsymbol{\Sigma}={\bf\Lambda}_{1}=diag(\alpha_{1},\underbrace{1,1,\dots,1}_{p-1})$.

  * $H_{2}$:

$\boldsymbol{\Sigma}={\bf\Lambda}_{2}=diag(\alpha_{1},\alpha_{2},\underbrace{1,1,\dots,1}_{p-2}),\alpha_{2}=0.9\alpha_{1}$.

  * $H_{3}$:

$\boldsymbol{\Sigma}={\bf\Lambda}_{3}=diag(\alpha_{1},\dots,\alpha_{5},\underbrace{1,1,\dots,1}_{p-5}),\alpha_{2}=0.9\alpha_{1},\alpha_{3}=0.85\alpha_{1},\alpha_{4}=0.8\alpha_{1},\alpha_{5}=0.75\alpha_{1}.$

  * $H_{4}$:

$\boldsymbol{\Sigma}={\bf U}_{0}{\bf\Lambda}_{1}{\bf U}_{0}^{*},$ where ${\bf
U}_{0}$ is the left singular vectors of a $p\times p$ random matrix with
i.i.d. $N(0,1)$ entries

  * $H_{5}$:

$\boldsymbol{\Sigma}={\bf U}_{0}{\bf\Lambda}_{2}{\bf U}_{0}^{*},$ and ${\bf
U}_{0}$ is defined in $H_{4}.$

  * $H_{6}$:

$\boldsymbol{\Sigma}={\bf U}_{0}{\bf\Lambda}_{3}{\bf U}_{0}^{*},$ and ${\bf
U}_{0}$ is defined in $H_{4}.$

Note that in above settings, $\boldsymbol{\Sigma}$ is diagonal and
$\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}=1$ under $H_{1}$–$H_{3}$, whereas
$\boldsymbol{\Sigma}$ is nondiagonal and
$\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}\asymp 1/p$ under $H_{4}$–$H_{6}$.

The settings for the significance level $\xi$ are constructed as follows:
$0.05,$ $0.01,$ and $1\times 10^{-4}.$ The empirical results are obtained
based on 10,000 replications with dimension $p=50,100,200$, respectively. We
set the RDS $p/n=1/3$.

In Tables 3–6, we list the empirical sizes and powers of CLRT, CNTT, and RLRT
under different settings. In the captions of these tables,
$``(Dt_{\ast},H_{\ast})"$ stands for the setting $Dt_{\ast},H_{\ast}$. For the
alternative hypothesis, due to space limitations, we present only some
selected tables with significant properties in the paper, and the tables for
other cases are provided in the supplementary material. Below are our
conclusions based on our simulation studies:

  1. (1)

For the null hypothesis, the empirical sizes of CLRT and CNTT are closer to
significance levels than that of RLRT overall. This is because the rate of
convergence for the largest eigenvalue distributions is slow. This property
has been discussed extensively in the literature and we omit the details here.
We refer interested readers to Johnstone (2001, 2008).

  2. (2)

For the alternative hypothesis:

     * •

From Table 4, it is easy to find that RLRT has the highest asymptotic power
under $H_{1}$. When there are two spikes, as seen from Table 5, CNTT and RLRT
have higher asymptotic power than CLRT. As seen from Table 6, when there are
five spikes, CNTT seems to have the highest asymptotic empirical power. This
is consistent with our analysis in subsection 4.2 that when the number of
spikes increases, CLRT and CNTT may exhibit higher asymptotic power than RLRT
in some scenarios. To be noticed that, in Table 6, we only list the powers
when $\xi=1\times 10^{-4}$ and consider smaller values of $\alpha_{1}$ since
the powers of three tests are all equal to 1 when $\xi=0.05$ and $\xi=0.01,$
or when $\alpha_{1}$ is not small enough, making comparisons infeasible.

     * •

From Tables 4–6, we can also find that in each row, the asymptotic power
derived under $(Dt_{2},H_{\ast})$ is smaller than that under
$(Dt_{1},H_{\ast})$, and power derived under $(Dt_{3},H_{\ast})$ is larger
than that under $(Dt_{1},H_{\ast}).$ This is because when
$\boldsymbol{\Sigma}$ is diagonal, a smaller $\beta_{x}$ may result in higher
asymptotic power, corresponding to Theorems 4.3 and 4.4.

Table 3: Empirical probability of rejecting $H_{0}$ at significance levels $0.05$ and $0.01$ under assumptions of Gaussian, Gamma, and Uniform distributions |  | $Dt_{1}$ | $Dt_{2}$ | $Dt_{3}$
---|---|---|---|---
test |  (p,n) | $\xi=0.05$ | $\xi=0.01$ | $\xi=0.05$ | $\xi=0.01$ | $\xi=0.05$ | $\xi=0.01$
$CLRT$ | (50,150) | 0.0563 | 0.0132 | 0.0545 | 0.0123 | 0.0538 | 0.0128
| (100,300) | 0.0517 | 0.0105 | 0.0567 | 0.0120 | 0.0525 | 0.0102
| (200,600) | 0.0543 | 0.0117 | 0.0502 | 0.0110 | 0.0527 | 0.0101
$CNTT$ | (50,150) | 0.0591 | 0.0138 | 0.0784 | 0.0219 | 0.0522 | 0.0107
| (100,300) | 0.0530 | 0.0106 | 0.0633 | 0.0175 | 0.0513 | 0.0112
| (200,600) | 0.0530 | 0.0122 | 0.0542 | 0.0142 | 0.0493 | 0.0117
$RLRT$ | (50,150) | 0.0426 | 0.0089 | 0.1104 | 0.0328 | 0.0215 | 0.0033
| (100,300) | 0.0459 | 0.0082 | 0.0969 | 0.0260 | 0.0276 | 0.0046
| (200,600) | 0.0495 | 0.0095 | 0.0779 | 0.0170 | 0.0279 | 0.0068
Table 4: Empirical probability of rejecting $H_{1}$ at significance level $\xi=0.05$ under assumptions of Gaussian, Gamma, and Uniform distributions  |  |  ($Dt_{1},H_{1}$) |  ($Dt_{2},H_{1}$) |  ($Dt_{3},H_{1}$)
---|---|---|---|---
test |  (p,n) | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$ | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$ | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$
$CLRT$ | (50,150) | 0.7236 | 0.9989 | 1 | 0.6920 | 0.9981 | 1 | 0.7511 | 1 | 1
| (100,300) | 0.7439 | 0.9999 | 1 | 0.7251 | 0.9998 | 1 | 0.7499 | 1 | 1
| (200,600) | 0.7544 | 0.9999 | 1 | 0.7468 | 0.9998 | 1 | 0.7632 | 1 | 1
$CNTT$ | (50,150) | 0.9919 | 1 | 1 | 0.9702 | 1 | 1 | 0.9996 | 1 | 1
| (100,300) | 0.9990 | 1 | 1 | 0.9898 | 1 | 1 | 1 | 1 | 1
| (200,600) | 0.9998 | 1 | 1 | 0.9973 | 1 | 1 | 1 | 1 | 1
$RLRT$ | (50,150) | 0.9998 | 1 | 1 | 0.9988 | 1 | 1 | 1 | 1 | 1
| (100,300) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| (200,600) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
Table 5: Empirical probability of rejecting $H_{2}$ at significance level $\xi=0.05$ under assumptions of Gaussian, Gamma, and Uniform distributions  |  |  ($Dt_{1},H_{2}$) |  ($Dt_{2},H_{2}$) |  ($Dt_{3},H_{2}$)
---|---|---|---|---
test |  (p,n) | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$ | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$ | $\alpha_{1}=3$ | $\alpha_{1}=5$ | $\alpha_{1}=7$
$CLRT$ | (50,150) | 0.9811 | 1 | 1 | 0.9633 | 1 | 1 | 0.9899 | 1 | 1
| (100,300) | 0.9894 | 1 | 1 | 0.9835 | 1 | 1 | 0.9929 | 1 | 1
| (200,600) | 0.9928 | 1 | 1 | 0.9894 | 1 | 1 | 0.9948 | 1 | 1
$CNTT$ | (50,150) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| (100,300) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| (200,600) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$RLRT$ | (50,150) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| (100,300) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| (200,600) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
Table 6: Empirical probability of rejecting $H_{3}$ at significance level $\xi=1\times 10^{-4}$ under assumptions of Gaussian, Gamma, and Uniform distributions  |  |  ($Dt_{1},H_{3}$) |  ($Dt_{2},H_{3}$) |  ($Dt_{3},H_{3}$)
---|---|---|---|---
|  |  $\alpha_{1}$
test |  (p,n) | $2.2$ | $2.5$ | $2.8$ | $2.2$ | $2.5$ | $2.8$ | $2.2$ | $2.5$ | $2.8$
$CLRT$ | (50,150) | 0.3911 | 0.8636 | 0.9918 | 0.3883 | 0.8232 | 0.9834 | 0.3825 | 0.8945 | 0.9976
| (100,300) | 0.3876 | 0.8946 | 0.9985 | 0.3804 | 0.8705 | 0.9951 | 0.3861 | 0.9110 | 0.9985
| (200,600) | 0.3779 | 0.9032 | 0.9980 | 0.3846 | 0.8946 | 0.9976 | 0.3825 | 0.9134 | 0.9997
$CNTT$ | (50,150) | 0.9972 | 1 | 1 | 0.9972 | 0.9998 | 1 | 0.9999 | 1 | 1
| (100,300) | 0.9997 | 1 | 1 | 0.9917 | 1 | 1 | 1 | 1 | 1
| (200,600) | 1 | 1 | 1 | 0.9964 | 1 | 1 | 1 | 1 | 1
$RLRT$ | (50,150) | 0.8933 | 0.9968 | 1 | 0.8954 | 0.9943 | 0.9999 | 0.8909 | 0.9994 | 1
| (100,300) | 0.9821 | 1 | 1 | 0.9756 | 0.9999 | 1 | 0.9904 | 1 | 1
| (200,600) | 0.9996 | 1 | 1 | 0.9989 | 1 | 1 | 1 | 1 | 1

##  6 Technical proofs

In this section, we present some lemmas that are needed in the proofs of the
main results. The truncation and renormalization are postponed to the end of
this paper.

###  6.1 Some primary definitions and lemmas

In this section, we provide some useful results that are used later in the
proofs of Theorems 3.1 and 3.2. For the population covariance matrix
$\boldsymbol{\Sigma}={\bf T}{\bf T}^{\ast}$, we consider the corresponding
sample covariance matrix ${\bf B}={\bf T}{\bf S}_{x}{\bf T}^{\ast}$, where
${\bf S}_{x}=\frac{1}{n}{\bf X}{\bf X}^{\ast}$. By singular value
decomposition of ${\bf T}$ (see (2.7)),

${\bf B}={\bf V}\left(\begin{array}[]{cc}{\bf D}_{1}^{\frac{1}{2}}{\bf
U}_{1}^{\ast}{\bf S}_{x}{\bf U}_{1}{\bf D}_{1}^{\frac{1}{2}},&{\bf
D}_{1}^{\frac{1}{2}}{\bf U}_{1}^{\ast}{\bf S}_{x}{\bf U}_{2}{\bf
D}_{2}^{\frac{1}{2}}\\\ {\bf D}_{2}^{\frac{1}{2}}{\bf U}_{2}^{\ast}{\bf
S}_{x}{\bf U}_{1}{\bf D}_{1}^{\frac{1}{2}},&{\bf D}_{2}^{\frac{1}{2}}{\bf
U}_{2}^{\ast}{\bf S}_{x}{\bf U}_{2}{\bf
D}_{2}^{\frac{1}{2}}\end{array}\right){\bf V}^{\ast}.$

Note that

${\bf S}=\left(\begin{array}[]{cc}{\bf D}_{1}^{\frac{1}{2}}{\bf
U}_{1}^{\ast}{\bf S}_{x}{\bf U}_{1}{\bf D}_{1}^{\frac{1}{2}},&{\bf
D}_{1}^{\frac{1}{2}}{\bf U}_{1}^{\ast}{\bf S}_{x}{\bf U}_{2}{\bf
D}_{2}^{\frac{1}{2}}\\\ {\bf D}_{2}^{\frac{1}{2}}{\bf U}_{2}^{\ast}{\bf
S}_{x}{\bf U}_{1}{\bf D}_{1}^{\frac{1}{2}},&{\bf D}_{2}^{\frac{1}{2}}{\bf
U}_{2}^{\ast}{\bf S}_{x}{\bf U}_{2}{\bf
D}_{2}^{\frac{1}{2}}\end{array}\right)\triangleq\left(\begin{array}[]{cc}{\bf
S}_{11},&{\bf S}_{12}\\\ {\bf S}_{21},&{\bf S}_{22}\end{array}\right).$

Moreover, ${\bf B}$ and ${\bf S}$ have the same eigenvalues.

Recall that $\underline{{\bf B}}=\frac{1}{n}{\bf X}^{\ast}{\bf T}^{\ast}{\bf
T}{\bf X}$ (the spectrum of which differs from that of ${\bf B}$ by
$\left|n-p\right|$ zeros). Its LSD is $\underline{F}^{c,H}$,
$\underline{F}^{c,H}\equiv\left(1-c\right)\mathbbm{1}_{\left[0,\infty\right)}+cF^{c,H}$,
and its Stieltjes transform is $\underline{m}\left(z\right)$. Let
$\widetilde{{\bf\lambda}}_{j}$ be the eigenvalues of ${\bf S}_{22}$ so that
the LSS of ${\bf S}_{22}$ is
$\sum_{j=1}^{p-M}f(\widetilde{{\bf\lambda}}_{j})$. Correspondingly, recall
that $c_{nM}:=\frac{p-M}{n}$, $H_{2n}:=F^{{\bf D}_{2}}$ and
$m_{2n0}:=m_{2n0}(z)$ is the Stieltjes transform of $F^{c_{nM},H_{2n}}$.
First, in Lemma 6.1 we derive that the difference between the two centers is
$0$.

######  Lemma 6.1.

Under Assumptions 1–4,

$\left(p-M\right)\int f\left(x\right)dF^{c_{nM},H_{2n}}=p\int
f\left(x\right)dF^{c_{n},H_{n}}\left(x\right).$

###### Proof.

By the Cauchy integral formula,

$p\int f(x)dF^{c_{n},H_{n}}=-\frac{p}{2\pi
i}\oint_{\mathcal{C}}f(z)m_{1n0}dz=-\frac{n}{2\pi
i}\oint_{\mathcal{C}}f(z)\underline{m}_{1n0}dz,$ $(p-M)\int
f(x)dF^{c_{nM},H_{2n}}=-\frac{p-M}{2\pi
i}\oint_{\mathcal{C}}f(z)m_{2n0}dz=-\frac{n}{2\pi
i}\oint_{\mathcal{C}}f(z)\underline{m}_{2n0}dz,$

where $\underline{m}_{1n0}$ and $\underline{m}_{2n0}$ are the Stieltjes
transforms of $\underline{F}^{c_{n},H_{n}}$ and
$\underline{F}^{c_{nM},H_{2n}}$, respectively. Then,

$(p-M)\int f(x)dF^{c_{nM},H_{2n}}-p\int f(x)dF^{c_{n},H_{n}}=\frac{n}{2\pi
i}\oint_{\mathcal{C}}f(z)\left(\underline{m}_{1n0}-\underline{m}_{2n0}\right)dz.$

Next, we prove that $\underline{m}_{1n0}=\underline{m}_{2n0}$.

Note that $m_{1n0}$ and $m_{2n0}$ are the unique solutions to

$\displaystyle
z=-\frac{1}{\underline{m}_{1n0}}+c_{n}\int\frac{tdH_{n}\left(t\right)}{1+t\underline{m}_{1n0}}$
(6.17) $\displaystyle
z=-\frac{1}{\underline{m}_{2n0}}+c_{nM}\int\frac{tdH_{2n}\left(t\right)}{1+t\underline{m}_{2n0}},$
(6.18)

respectively, where $\underline{m}_{1n0}=-\frac{1-c_{n}}{z}+c_{n}m_{1n0}$ and
$\underline{m}_{2n0}=-\frac{1-c_{nM}}{z}+c_{nM}m_{2n0}$. Since

$H_{n}(t)=\frac{1}{p}\left[\sum_{i=1}^{M}\mathbbm{1}_{\left\\{0\leq
t\right\\}}+\sum_{i=M+1}^{p}\mathbbm{1}_{\left\\{\alpha_{i}\leq
t\right\\}}\right]=\frac{M}{p}+\frac{1}{p}\sum_{i=M+1}^{p}\mathbbm{1}_{\left\\{\alpha_{i}\leq
t\right\\}}$

and
$H_{2n}(t)=\frac{1}{p-M}\sum_{i=M+1}^{p}\mathbbm{1}_{\left\\{\alpha_{i}\leq
t\right\\}}$, (6.17) can be written as

$\displaystyle z$
$\displaystyle=-\frac{1}{\underline{m}_{1n0}}+\frac{p}{n}\int\frac{td\left(\frac{M}{p}+\frac{1}{p}\sum_{i=M+1}^{p}\mathbbm{1}_{\left\\{\alpha_{i}\leq
t\right\\}}\left(t\right)\right)}{1+t\underline{m}_{1n0}}=-\frac{1}{\underline{m}_{1n0}}+\frac{p}{n}\int\frac{td\left(\frac{1}{p}\sum_{i=M+1}^{p}\mathbbm{1}_{\left\\{\alpha_{i}\leq
t\right\\}}\left(t\right)\right)}{1+t\underline{m}_{1n0}}$
$\displaystyle=-\frac{1}{\underline{m}_{1n0}}+\frac{1}{n}\sum_{i=M+1}^{p}\frac{\alpha_{i}}{1+\alpha_{i}\underline{m}_{1n0}}.$
(6.19)

Similarly, equation (6.18) can be written as
$z=-\frac{1}{\underline{m}_{2n0}}+\frac{1}{n}\sum_{i=M+1}^{p}\frac{\alpha_{i}}{1+\alpha_{i}\underline{m}_{2n0}}.$
Thus, according to the fact that $m_{1n0}$ and $m_{2n0}$ are the unique
solutions of (6.19) and (6.1), respectively, we have $m_{1n0}=m_{2n0}$, which
completes the proof of this lemma.

∎

Note that for the bounded part of the LSS,
$\sum_{j=M+1}^{p}f\left(\lambda_{j}\right)$, the BST cannot be used directly
since it is not an LSS of a sample covariance matrix. In fact, it approximates
the LSS of ${\bf S}_{22}$, that is
$\sum_{j=1}^{p-M}f\left(\tilde{\lambda}_{j}\right)$, but they are not equal
since the off-diagonal blocks of the sample covariance matrix are not null.
The following lemma measures the difference between
$\sum_{j=M+1}^{p}f\left(\lambda_{j}\right)$ and
$\sum_{j=1}^{p-M}f\left(\tilde{\lambda}_{j}\right)$.

######  Lemma 6.2.

Under Assumptions 1–4,

$\sum_{j=M+1}^{p}f\left(\lambda_{j}\right)-\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)\frac{\underline{m}_{2n0}^{{}^{\prime}}(z)}{\underline{m}_{2n0}(z)}dz=o_{p}(1).$

###### Proof.

Note that $L_{1}:=\sum_{j=M+1}^{p}f\left(\lambda_{j}\right).$ By the Cauchy
integral formula, we have $L_{1}=-\frac{p}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)m_{n}\left(z\right)dz,$ where
$m_{n}=\frac{1}{p}\mathrm{tr}\left({\bf S}-z{\bf
I}_{p}\right)^{-1}=\frac{1}{p}\mathrm{tr}\left({\bf B}-z{\bf
I}_{p}\right)^{-1}$. Analogously, we have
$L_{2}:=\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)=-\frac{p-M}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)m_{2n}\left(z\right)dz,$ where
$m_{2n}=\frac{1}{p-M}\mathrm{tr}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-1}$. By applying the block matrix inversion formula to
$m_{n}$, we can obtain

$\displaystyle L_{1}-L_{2}=-\frac{1}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)\left(T_{1}-T_{2}\right)dz,$ (6.20)

where

$\displaystyle T_{1}$ $\displaystyle=\mathrm{tr}\left({\bf S}_{11}-z{\bf
I}_{M}-{\bf S}_{12}\left({\bf S}_{22}-z{\bf I}_{p-M}\right)^{-1}{\bf
S}_{21}\right)^{-1},$ $\displaystyle T_{2}$
$\displaystyle=-\mathrm{tr}\left[\left({\bf S}_{11}-z{\bf I}_{M}-{\bf
S}_{12}\left({\bf S}_{22}-z{\bf I}_{p-M}\right)^{-1}{\bf
S}_{21}\right)^{-1}{\bf S}_{12}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-2}{\bf S}_{21}\right].$

Note that for any matrix ${\bf Z}$,

$\displaystyle{\bf Z}\left({\bf Z}^{\ast}{\bf Z}-\lambda{\bf
I}\right)^{-1}{\bf Z}^{\ast}={\bf I}+\lambda\left({\bf Z}{\bf
Z}^{\ast}-\lambda{\bf I}\right)^{-1},$

which, together with the notation ${\bf\Upsilon}_{n}:=\frac{1}{n}{\bf
D}_{1}^{\frac{1}{2}}{\bf U}_{1}^{\ast}{\bf X}\left(\frac{1}{n}{\bf
X}^{\ast}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf X}-z{\bf
I}_{n}\right)^{-1}{\bf X}^{\ast}{\bf U}_{1}{\bf D}_{1}^{\frac{1}{2}}$, implies
that

$\displaystyle T_{1}$ $\displaystyle=-z^{-1}\mathrm{tr}\left({\bf
I}_{M}+{\bf\Upsilon}_{n}\right)^{-1}$ $\displaystyle T_{2}$
$\displaystyle=z^{-1}\mathrm{tr}\left[\left({\bf
I}_{M}+{\bf\Upsilon}_{n}\right)^{-1}{\bf S}_{12}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-2}{\bf S}_{21}\right]$

$\underline{m}_{2n}=\underline{m}_{2n}(z)$ denotes the Stieltjes transform of
$F^{\frac{1}{n}{\bf X}^{\ast}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf X}}$.
Thus, we have that $\underline{m}_{2n}(z)-\underline{m}(z)=o_{p}(1)$ for any
$z\in\mathcal{C}$. From Theorem 3.1 of (Jiang and Bai, 2021), we know that

$\displaystyle\frac{1}{n}{\bf U}_{1}^{\ast}{\bf X}\left(\frac{1}{n}{\bf
X}^{\ast}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf X}-z{\bf
I}_{n}\right)^{-1}{\bf X}^{\ast}{\bf
U}_{1}=\underline{m}_{2n}\left(z\right){\bf I}_{M}+O_{p}(n^{-\frac{1}{2}}).$
(6.21)

Thus, under Assumption 3, we find that

$\displaystyle{\bf D}_{1}^{1/2}\left({\bf
I}_{M}+{\bf\Upsilon}_{n}\right)^{-1}{\bf
D}_{1}^{1/2}=\frac{1}{\underline{m}\left(z\right)}{\bf I}_{M}+o_{p}(1),$
(6.22)

which yields

$\displaystyle T_{1}=o_{p}(1).$ (6.23)

It follows that

$\displaystyle{\bf D}_{1}^{-1/2}{\bf S}_{12}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-2}{\bf S}_{21}{\bf D}_{1}^{-1/2}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\mathrm{tr}\left[\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-2}{\bf S}_{22}\right]{\bf I}_{M}+O_{p}(n^{-\frac{1}{2}})$
$\displaystyle=$ $\displaystyle\frac{1}{n}\mathrm{tr}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-1}{\bf I}_{M}+\frac{z}{n}\mathrm{tr}\left({\bf S}_{22}-z{\bf
I}_{p-M}\right)^{-2}{\bf I}_{M}+O_{p}(n^{-\frac{1}{2}})$ $\displaystyle=$
$\displaystyle cm_{2n}\left(z\right){\bf
I}_{M}+zcm_{2n}^{\prime}\left(z\right){\bf I}_{M}+o_{p}(1)$ $\displaystyle=$
$\displaystyle cm_{2n0}\left(z\right){\bf
I}_{M}+zcm_{2n0}^{\prime}\left(z\right){\bf I}_{M}+o_{p}(1)$ $\displaystyle=$
$\displaystyle\underline{m}_{2n0}\left(z\right){\bf
I}_{M}+z\underline{m}_{2n0}^{\prime}\left(z\right){\bf I}_{M}+o_{p}(1),$
(6.24)

where the last equality is derived from
$\underline{m}_{2n0}(z)=-\frac{1-c}{z}+cm_{2n0}\left(z\right)$. Therefore,
according to (6.22) and (6.24), we obtain

$\displaystyle
T_{2}=M\frac{\underline{m}_{2n0}\left(z\right)+z\underline{m}_{2n0}^{\prime}\left(z\right)}{z\underline{m}_{2n0}\left(z\right)}+o_{p}(1),$

which, together with (6.20) and (6.23), implies that

$\displaystyle L_{1}-L_{2}=\frac{M}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)\frac{\underline{m}_{2n0}(z)+z\underline{m}_{2n0}^{{}^{\prime}}(z)}{z\underline{m}_{2n0}(z)}dz+o_{p}(1)=\frac{M}{2\pi
i}\oint_{\mathcal{C}}f\left(z\right)\frac{\underline{m}_{2n0}^{{}^{\prime}}(z)}{\underline{m}_{2n0}(z)}dz+o_{p}(1).$

Therefore, the proof of this lemma is complete. ∎

Define random vector
$\bm{\gamma}_{k}=\left(\gamma_{kj}\right)^{\top}=\left(\sqrt{n}\frac{\lambda_{j}-\phi_{n}\left(\alpha_{k}\right)}{\phi_{n}\left(\alpha_{k}\right)},j\in
J_{k}\right)^{\top}$, where $J_{k}$ is the indicator set of a packet of
$d_{k}$ consecutive sample eigenvalues. Then, we present the following lemma,
which is borrowed from Jiang and Bai (2021) and characterizes the limiting
distribution of the spiked eigenvalues of the sample covariance matrix.

######  Lemma 6.3.

(Jiang and Bai (2021)) Under Assumptions 1–4, random vector $\bm{\gamma}_{k}$
converges weakly to the joint distribution of $d_{k}$ eigenvalues of a
Gaussian random matrix

$-\frac{1}{\theta_{k}}\left[{\bf\Omega}_{\phi_{k}}\right]_{kk},$

where

$\theta_{k}=\phi_{k}^{2}\underline{m}_{2}\left(\phi_{k}\right),~{}~{}\underline{m}_{2}\left(\lambda\right)=\int\frac{1}{\left(\lambda-x\right)^{2}}d\underline{F}^{c,H}\left(x\right)$

with $\underline{F}^{c,H}$ being the LSD of matrix $n^{-1}{\bf X}^{\ast}{\bf
U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf X},$
$\phi_{k}=\alpha_{k}\left(1+c\int\frac{t}{\alpha_{k}-t}dH\left(t\right)\right)$.
$\left[{\bf\Omega}_{\phi_{k}}\right]_{kk}$ is the $k$th diagonal block of
matrix ${\bf\Omega}_{\phi_{k}}$. The variances and covariances of the elements
$\omega_{ij}$ of ${\bf\Omega}_{\phi_{k}}$ are:

$\operatorname{Cov}\left(\omega_{i_{1},j_{1}},\omega_{i_{2},j_{2}}\right)=\left\\{\begin{array}[]{cc}(\alpha_{x}+1)\theta_{k}+\beta_{x}\mathcal{U}_{iiii}\nu_{k},&i_{1}=j_{1}=i_{2}=j_{2}=i\\\
\theta_{k}+\beta_{x}\mathcal{U}_{ijij}\nu_{k},&i_{1}=i_{2}=i\neq
j_{1}=j_{2}=j\\\ \beta_{x}\mathcal{U}_{i_{1}j_{1}i_{2}j_{2}}\nu_{k},&\text{
other cases }\end{array}\right.$

where
$\beta_{x}\mathcal{U}_{i_{1}j_{1}i_{2}j_{2}}=\sum_{t=1}^{p}\bar{u}_{ti_{1}}u_{tj_{1}}u_{ti_{2}}\bar{u}_{tj_{2}}\beta_{x}$,
$\bm{u}_{i}=\left(u_{1i},\ldots,u_{pi}\right)^{\top}$ are the $i$ th column of
the matrix $\mathbf{U}_{1}$,
$\nu_{k}=\phi_{k}^{2}\underline{m}^{2}\left(\phi_{k}\right)$.

Recall that $\lambda_{j}$ are the eigenvalues of ${\bf B}$, and
$\widetilde{{\bf\lambda}}_{j}$ are the eigenvalues of ${\bf S}_{22}$. The
following lemma shows the asymptotic independence between
$\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ and
$\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)$.

######  Lemma 6.4.

Under Assumptions 1–4, $\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ and
$\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)$ are
asymptotically independent.

###### Proof.

It is sufficient to prove that for a given
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$, the asymptotic
limiting distribution of $\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ does not
depend on the random part of
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$, that is, it only
depends on its limit.

First, we consider $f(x)=x$. From the proof of Theorem 3.1 in Jiang and Bai
(2021), we have the following determinant equation

$0=\left|\left[{\bf\Omega}_{M}\left(\phi_{k}\right)\right]_{kk}+\mathrm{lim}\gamma_{kj}\left\\{\phi_{k}^{2}\underline{m}_{2}(\phi_{k})\right\\}{\bf
I}_{d_{k}}\right|,$

where ${\bf\Omega}_{M}\left(\phi_{k}\right)$

$\displaystyle=\frac{\phi_{k}}{\sqrt{n}}\left[\mathrm{tr}\left\\{\left(\phi_{k}{\bf
I}-\frac{1}{n}{\bf X}^{*}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf
X}\right)^{-1}\right\\}{\bf I}-{\bf U}_{1}^{*}{\bf X}\left(\phi_{k}{\bf
I}-\frac{1}{n}{\bf X}^{*}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf
X}\right)^{-1}{\bf X}^{*}{\bf U}_{1}\right],$

and $\underline{m}_{2}(\phi_{k})$ is the limit of
$\mathrm{tr}\left(\phi_{k}{\bf I}-\frac{1}{n}{\bf X}^{*}{\bf U}_{2}{\bf
D}_{2}{\bf U}_{2}^{*}{\bf X}\right)^{-2}$. Then, we know that $\gamma_{kj}$
has the same asymptotic distribution with eigenvalues of
$-\frac{\left[{\bf\Omega}_{M}\left(\phi_{k}\right)\right]_{kk}}{\phi_{k}^{2}\underline{m}_{2}(\phi_{k})}$
in order. From Jiang and Bai (2021), we can obtain that the limiting
distribution of $\bm{\Omega}_{M}\left(\phi_{k}\right)$ does not change if
${\bf U}_{2}^{*}{\bf X}$ is replaced by ${\bf U}_{2}^{*}{\bf Y}$ while ${\bf
U}_{1}^{*}{\bf X}$ remains unchanged. Here ${\bf Y}$ and ${\bf X}$ are i.i.d..
Therefore in $\bm{\Omega}_{M}\left(\phi_{k}\right)$, we can assume that ${\bf
U}_{1}^{*}{\bf X}$ and ${\bf U}_{2}^{*}{\bf X}$ are independent without loss
of generality. Then, given ${\bf U}_{2}^{*}{\bf X}$, the limiting distribution
of $\gamma_{kj}$ only depends on the limit of $\mathrm{tr}\left(\phi_{k}{\bf
I}-\frac{1}{n}{\bf X}^{*}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}{\bf
X}\right)^{-1}$, that is, $\underline{m}_{2}(\phi_{k})$, and has nothing to do
with the random part. Therefore, it is found that
$\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ and
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$ are asymptotically
independent when $f(x)=x.$

When $f(x)\neq x$, by using the Newton-Leibniz formula, we have

$\displaystyle\sum_{j=1}^{M}f\left(\lambda_{j}\right)-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)=\sum_{k=1}^{K}\sum_{j\in
J_{k}}(f\left(\lambda_{j}\right)-f\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right))$
$\displaystyle=\sum_{k=1}^{K}\sum_{j\in
J_{k}}\int_{0}^{\frac{\phi_{n}\left({\bf\alpha}_{k}\right)}{\sqrt{n}}\gamma_{kj}}f^{\prime}\left(t+\phi_{n}\left({\bf\alpha}_{k}\right)\right)dt$
$\displaystyle=\sum_{k=1}^{K}\sum_{j\in
J_{k}}\int_{0}^{1}\frac{\phi_{n}\left({\bf\alpha}_{k}\right)}{\sqrt{n}}\gamma_{kj}\frac{f^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\left(1+\frac{\gamma_{kj}}{\sqrt{n}}s\right)\right)}{f^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)}f^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)ds$
$\displaystyle=\sum_{k=1}^{K}\sum_{j\in
J_{k}}\int_{0}^{1}\varpi_{nk}\gamma_{kj}\frac{f^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\left(1+\frac{\gamma_{kj}}{\sqrt{n}}s\right)\right)}{f^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)}ds$
$\displaystyle\rightarrow\sum_{k=1}^{K}\sum_{j\in
J_{k}}\int_{0}^{1}\gamma_{kj}\varpi_{nk}ds=\sum_{k=1}^{K}\sum_{j\in
J_{k}}\varpi_{nk}\gamma_{kj},$ (6.25)

where (6.25) is true due to Assumption 4, and we denote
$\varpi_{nk}=\dfrac{\phi_{n}(\alpha_{k})}{\sqrt{n}}f^{\prime}(\phi_{n}(\alpha_{k}))$.
Thus, we convert the above equation into a function of $\gamma_{kj}$. The
above calculations represent the underlying idea of the generalized delta
method we mentioned in the Introduction. Since we have proven above that given
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$, the limiting
distribution of $\gamma_{kj}$ is concerned only with the limit of
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$, as is
$\sum_{k=1}^{K}\sum_{j\in J_{k}}\varpi_{nk}\gamma_{kj}$, accordingly, we can
conclude that $\sum_{j=1}^{M}f\left(\lambda_{j}\right)$ and
$\sum_{j=1}^{p-M}f\left(\widetilde{\lambda}_{j}\right)$ are asymptotically
independent. The proof is complete. ∎

In the following lemma, we derive the asymptotic distribution of the LSS
generated from submatrix ${\bf S}_{22}$.

######  Lemma 6.5.

Define $Q_{1}=\sum_{j=1}^{p-M}f_{1}(\widetilde{\lambda}_{j})-(p-M)\int
f_{1}(x)dF^{c_{nM},H_{2n}}$; then, under Assumptions 1–4, we have

$\kappa_{1}^{-1}\left(Q_{1}-\mu_{1}\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N\left(0,1\right)$

with mean function

$\displaystyle\mu_{1}$ $\displaystyle=-\frac{\alpha_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}f_{1}(z)\frac{c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{\left(1-c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)\left(1-\alpha_{x}c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)}dz$
$\displaystyle-\frac{\beta_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}f_{1}(z)\frac{c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{1-c_{nM}\int\underline{m}_{2n0}^{2}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-2}dH_{2n}(t)}dz,$

and the covariance function is

$\displaystyle\kappa_{1}^{2}=-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{1}\left(z_{1}\right)f_{1}\left(z_{2}\right)\vartheta_{n}^{2}dz_{1}dz_{2},$

where
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}),$

$\displaystyle\Theta_{0,n}(z_{1},z_{2})$
$\displaystyle=\dfrac{\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{(\underline{m}_{2n0}(z_{1})-\underline{m}_{2n0}(z_{2}))^{2}}-\dfrac{1}{(z_{1}-z_{2})^{2}},$
$\displaystyle\Theta_{1,n}(z_{1},z_{2})$
$\displaystyle=\frac{\partial}{\partial
z_{2}}\left\\{\dfrac{\partial\mathcal{A}_{n}(z_{1},z_{2})}{\partial
z_{1}}\dfrac{1}{1-\alpha_{x}\mathcal{A}_{n}(z_{1},z_{2})}\right\\},$
$\displaystyle\mathcal{A}_{n}(z_{1},z_{2})$
$\displaystyle=\dfrac{z_{1}z_{2}}{n}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\mathrm{tr}{{\bf\Gamma}^{*}{\bf
P}_{n}(z_{1}){\bf\Gamma}{\bf\Gamma}^{\top}{\bf
P}_{n}(z_{2})^{\top}\bar{{\bf\Gamma}}},$
$\displaystyle\Theta_{2,n}(z_{1},z_{2})$
$\displaystyle=\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii},$

and the definitions of ${\bf P}_{n}$, ${\bf\Gamma}$, and $\underline{m}_{2n0}$
are defined in Section 3.

###### Proof.

From Zheng et al. (2015), we have that under Assumptions 1–4, the random
variable
$\left(\kappa_{1}^{0}\right)^{-1}\left(Q_{1}-\mu_{1}\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N\left(0,1\right),$ with mean function

$\displaystyle\mu_{1}$ $\displaystyle=-\frac{\alpha_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}\frac{f_{1}(z)c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{\left(1-c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)\left(1-\alpha_{x}c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)}dz$
$\displaystyle-\frac{\beta_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}\frac{f_{1}(z)c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{1-c_{nM}\int\underline{m}_{2n0}^{2}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-2}dH_{2n}(t)}dz,$

and the covariance function is

$\displaystyle\left(\kappa_{1}^{0}\right)^{2}=$
$\displaystyle-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{1}\left(z_{1}\right)f_{1}\left(z_{2}\right)(\vartheta_{n}^{0})^{2}dz_{1}dz_{2},$

where

$\displaystyle(\vartheta_{n}^{0})^{2}=$
$\displaystyle\frac{b_{n}\left(z_{1}\right)b_{n}\left(z_{2}\right)}{n^{2}}\sum_{j=1}^{n}\operatorname{tr}\mathbb{E}_{j}{\bf\Gamma}{\bf\Gamma}^{*}\mathbf{A}_{j}^{-1}\left(z_{1}\right)\mathbb{E}_{j}\left({\bf\Gamma}{\bf\Gamma}^{*}\mathbf{A}_{j}^{-1}\left(z_{2}\right)\right)$
$\displaystyle+\frac{\alpha_{x}b_{n}\left(z_{1}\right)b_{n}\left(z_{2}\right)}{n^{2}}\sum_{j=1}^{n}\operatorname{tr}\mathbb{E}_{j}{\bf\Gamma}^{*}\mathbf{A}_{j}^{-1}\left(z_{1}\right){\bf\Gamma}\mathbb{E}_{j}\left({\bf\Gamma}^{\top}\left(\mathbf{A}_{j}^{\top}\right)^{-1}\left(z_{2}\right)\bar{{\bf\Gamma}}\right)$
$\displaystyle+\frac{\beta_{x}b_{n}\left(z_{1}\right)b_{n}\left(z_{2}\right)}{n^{2}}\sum_{j=1}^{n}\sum_{i=1}^{p}\bm{e}_{i}^{\top}{{\bf\Gamma}}^{*}\mathbf{A}_{j}^{-1}\left(z_{1}\right){{\bf\Gamma}}\bm{e}_{i}\cdot\bm{e}_{i}^{\top}{{\bf\Gamma}}^{*}\mathbf{A}_{j}^{-1}\left(z_{2}\right){{\bf\Gamma}}\bm{e}_{i},$

where
$b_{n}\left(z\right)=\frac{1}{1+n^{-1}\mathbb{E}{\textrm{tr}}{\bf\Gamma}{\bf\Gamma}^{*}{\bf
A}_{j}^{-1}\left(z\right)}.$ The symbols ${\bf A}_{j},\bm{e_{i}}$ are defined
in Section 2. Moreover Najim and Yao (2016) provided an estimation
$\vartheta_{n}^{2}$ for $(\vartheta_{n}^{0})^{2}$ and proved that
$(\vartheta_{n}^{0})^{2}$ is close to $\vartheta_{n}^{2}$ in the Lévy–Prohorov
distance, where
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}),$

$\displaystyle\Theta_{0,n}(z_{1},z_{2})$
$\displaystyle=\dfrac{\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{(\underline{m}_{2n0}(z_{1})-\underline{m}_{2n0}(z_{2}))^{2}}-\dfrac{1}{(z_{1}-z_{2})^{2}},$
$\displaystyle\Theta_{1,n}(z_{1},z_{2})$
$\displaystyle=\frac{\partial}{\partial
z_{2}}\left\\{\dfrac{\partial\mathcal{A}_{n}(z_{1},z_{2})}{\partial
z_{1}}\dfrac{1}{1-\alpha_{x}\mathcal{A}_{n}(z_{1},z_{2})}\right\\},$
$\displaystyle\mathcal{A}_{n}(z_{1},z_{2})$
$\displaystyle=\dfrac{z_{1}z_{2}}{n}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\mathrm{tr}{{\bf\Gamma}^{*}{\bf
P}_{n}(z_{1}){\bf\Gamma}{\bf\Gamma}^{\top}{\bf
P}_{n}(z_{2})^{\top}\bar{{\bf\Gamma}}},$
$\displaystyle\Theta_{2,n}(z_{1},z_{2})$
$\displaystyle=\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii},$

and the definitions of ${\bf P}_{n}$, ${\bf\Gamma}$, and $\underline{m}_{2n0}$
are given in Section 3. Notably, if ${\bf\Gamma}$ is not real, then the
convergence of $\Theta_{1,n}(z_{1},z_{2})$ is not guaranteed. However, if
${\bf\Gamma}$ and entries $x_{ij}$ are real, that is, $\alpha_{x}=1$, then it
can be easily proven that
$\Theta_{0,n}(z_{1},z_{2})=\Theta_{1,n}(z_{1},z_{2})$. Similarly, the
convergence of $\Theta_{2,n}(z_{1},z_{2})$ depends on the assumption that
${\bf\Gamma}^{*}{\bf\Gamma}$ is diagonal; thus, under Assumptions 1–4,
$\Theta_{1,n}(z_{1},z_{2})$ and $\Theta_{2,n}(z_{1},z_{2})$ may have no
limits.

Therefore, the covariance term $\left(\kappa_{1}^{0}\right)^{2}$ is estimable,
and the estimate is $\kappa_{1}^{2}$, with

$\kappa_{1}^{2}=-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{1}\left(z_{1}\right)f_{1}\left(z_{2}\right)\vartheta_{n}^{2}dz_{1}dz_{2}.$

Thus, the proof is complete. ∎

###  6.2  Truncation and renormalization

In this subsection, we truncate and renormalize the random variables to ensure
the existence of their higher-order moments.

Similar to Jiang and Bai (2021), we may select $\eta_{n}\rightarrow 0$ such
that $\eta_{n}^{-4}n^{2}P(|x_{ij}|\geq\eta_{n}\sqrt{n})\to 0$. Let
$\hat{x}_{ij}=x_{ij}\mathbbm{1}_{\left\\{\left|x_{ij}\right|<\eta_{n}\sqrt{n}\right\\}}$
and
$\tilde{x}_{ij}=\frac{\hat{x}_{ij}-\mathbbm{E}\hat{x}_{ij}}{\hat{\sigma}_{n}}$,
where
$\hat{\sigma}_{n}^{2}=\mathbbm{E}\left|\hat{x}_{ij}-\mathbbm{E}\hat{x}_{ij}\right|^{2}$.
Analogous to the discussion in Li and Bai (2015), the sequence
$\eta_{n}\rightarrow 0$ can be made arbitrarily slow, therefore, we may
require it satisfying $\eta_{n}n^{t}\to\infty$ for any fixed $t>0$.
Correspondingly, we define $\hat{{\bf B}}=\frac{1}{n}{\bf T}\hat{{\bf
X}}\hat{{\bf X}}^{\ast}{\bf T}^{\ast}$ and $\tilde{{\bf B}}=\frac{1}{n}{\bf
T}\tilde{{\bf X}}\tilde{{\bf X}}^{\ast}{\bf T}^{\ast}$, where $\hat{{\bf
X}}=(\hat{x}_{ij})$ and $\tilde{{\bf X}}=(\tilde{x}_{ij})$. $\hat{G}_{n}$ and
$\tilde{G}_{n}$ denote the analogs of $G_{n}$ with the matrix ${\bf B}$
replaced by $\hat{{\bf B}}$ and $\tilde{{\bf B}}$, respectively. Next, we
demonstrate that the limiting distribution of the LSS is unchanged when the
entries of $\mathbf{X}$ are replaced by the truncated and renormalized
entries.

From Supplement B in Jiang and Bai (2021), we have

$\displaystyle\mathrm{P}({\bf B}\neq\hat{{\bf
B}})\leq\sum_{i,j}\mathrm{P}\left(\left|x_{ij}\right|\geq\eta_{n}\sqrt{n}\right)\leq
np\mathrm{P}\left(\left|x_{11}\right|\geq\eta_{n}\sqrt{n}\right)\rightarrow
0,\quad\text{ as }n,p\rightarrow\infty.$

It follows from the definition of LSS and Lemma 6.2, for any $l=1,\ldots,h$,

$\displaystyle\int f_{l}(x)d\hat{G}_{n}-\int
f_{l}(x)d\tilde{G}_{n}(x)=\sum_{i=1}^{p}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf B}}}))$
$\displaystyle=\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))+\sum_{i=M+1}^{p}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf B}}}))$
$\displaystyle=\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))+\sum_{i=M+1}^{p}f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-\sum_{i=1}^{p-M}f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})+\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}}))$
$\displaystyle+\sum_{i=1}^{p-M}f_{l}(\lambda_{i}^{\tilde{{\bf
S}}_{22}})-\sum_{i=M+1}^{p}f_{l}(\lambda_{i}^{\tilde{{\bf B}}})$
$\displaystyle=\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))+\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}}))$
$\displaystyle+\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{l}\left(z\right)\frac{\hat{\underline{m}}_{2n0}^{{}^{\prime}}(z)}{\hat{\underline{m}}_{2n0}(z)}dz-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{l}\left(z\right)\frac{\tilde{\underline{m}}_{2n0}^{{}^{\prime}}(z)}{\tilde{\underline{m}}_{2n0}(z)}dz.$

Here $\hat{{\bf S}}_{22}$ and $\tilde{{\bf S}}_{22}$ denote the analogs of
${\bf S}_{22}$ with the matrix ${\bf X}$ replaced by $\hat{{\bf X}}$ and
$\tilde{{\bf X}}$, respectively. $\hat{\underline{m}}_{2n0}(z)$ and
$\tilde{\underline{m}}_{2n0}(z)$ denote the analogs of
$\underline{m}_{2n0}(z)$ with the matrix ${\bf X}$ replaced by $\hat{{\bf X}}$
and $\tilde{{\bf X}}$, respectively.

Therefore,
$\left|\hat{Y}_{l}-\tilde{Y}_{l}\right|=\left|\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))+\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}}))\right|\leq\\\
\left|\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))\right|+\left|\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}}))\right|.$

For $\left|\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}}))\right|,$ we have

$\displaystyle\left|\sum_{i=1}^{p-M}(f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf
S}}_{22}}))\right|\leq\sum_{i=1}^{p-M}\left|f_{l}(\lambda_{i}^{\hat{{\bf
S}}_{22}})-f_{l}(\lambda_{i}^{\tilde{{\bf S}}_{22}})\right|\leq
C\sum_{i=1}^{p-M}\left|\lambda_{i}^{\hat{{\bf
S}}_{22}}-\lambda_{i}^{\tilde{{\bf S}}_{22}}\right|.$

Similar to Lemma 2.7 in Bai (1999), we have

$\displaystyle\sum_{i=1}^{p-M}\left|\lambda_{i}^{\hat{{\bf
S}}_{22}}-\lambda_{i}^{\tilde{{\bf
S}}_{22}}\right|\leq\left(\frac{2}{n}\mathrm{tr}(\hat{{\bf
S}}_{22}+\tilde{{\bf S}}_{22})\mathrm{tr}{\bf D}_{2}^{\frac{1}{2}}{\bf
U}_{2}^{*}(\hat{{\bf X}}-\tilde{{\bf X}})(\hat{{\bf X}}-\tilde{{\bf
X}})^{*}{\bf U}_{2}{\bf D}_{2}^{\frac{1}{2}}\right)^{1/2}$
$\displaystyle=\left(\frac{2}{n}\mathrm{tr}(\hat{{\bf S}}_{22}+\tilde{{\bf
S}}_{22})\mathrm{tr}(\hat{{\bf X}}-\tilde{{\bf X}})(\hat{{\bf X}}-\tilde{{\bf
X}})^{*}{\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}\right)^{1/2}$
$\displaystyle\leq\left(\frac{2C}{n}\mathrm{tr}(\hat{{\bf S}}_{22}+\tilde{{\bf
S}}_{22})\mathrm{tr}(\hat{{\bf X}}-\tilde{{\bf X}})(\hat{{\bf X}}-\tilde{{\bf
X}})^{*}\right)^{1/2},$

where the last inequality is because ${\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{*}$
is bounded. It is easy to prove $\frac{1}{n}\mathbb{E}\mathrm{tr}(\hat{{\bf
S}}_{22}+\tilde{{\bf S}}_{22})<C,$ and

$\displaystyle\mathbb{E}\mathrm{tr}(\hat{{\bf X}}-\tilde{{\bf X}})(\hat{{\bf
X}}-\tilde{{\bf
X}})^{*}=\sum_{i,j}\mathbb{E}\left|(1-\frac{1}{\hat{\sigma}_{n}})\hat{x}_{ij}+\frac{\mathbb{E}\hat{x}_{ij}}{\hat{\sigma}_{n}}\right|^{2}$
$\displaystyle=\sum_{i,j}\left(\mbox{var}((1-\frac{1}{\hat{\sigma}_{n}})\hat{x}_{ij}+\frac{\mathbb{E}\hat{x}_{ij}}{\hat{\sigma}_{n}})+\left|\mathbb{E}\hat{x}_{ij}\right|^{2}\right).$
(6.26)

Since
$\mbox{var}\left((1-\frac{1}{\hat{\sigma}_{n}})\hat{x}_{ij}+\frac{\mathbb{E}\hat{x}_{ij}}{\hat{\sigma}_{n}}\right)=(\hat{\sigma}_{n}-1)^{2}\leq\frac{(\hat{\sigma}_{n}^{2}-1)^{2}}{\hat{\sigma}_{n}^{2}+1}\leq
2\left(\mathbb{E}\left|x_{ij}\right|^{2}\mathbbm{1}_{\left\\{\left|x_{ij}\right|\geq\eta_{n}\sqrt{n}\right\\}}\right)^{2},$
and by the selection of $\eta_{n}$,
$\mathbb{E}\left|x_{ij}\right|^{2}\mathbbm{1}_{\left\\{\left|x_{ij}\right|\geq\eta_{n}\sqrt{n}\right\\}}=o(\eta_{n}^{2}n^{-1}),$
$\mathbb{E}\hat{x}_{ij}=o(\eta_{n}n^{-3/2}),$ then we can obtain (6.26) is
o(1). Therefore, $\sum_{i=1}^{p-M}\left|\lambda_{i}^{\hat{{\bf
S}}_{22}}-\lambda_{i}^{\tilde{{\bf S}}_{22}}\right|$ is $o_{p}(1)$.

For the first term $\left|\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf B}}}))\right|$, from the arguments in
Supplement B of Jiang and Bai (2021), we know that

$\displaystyle\left|\lambda_{i}^{\hat{{\bf B}}}-\lambda_{i}^{\tilde{{\bf
B}}}\right|=o_{p}(n^{-\frac{1}{2}}{\rho_{i}}).$ (6.27)

We recall that $\rho_{i}=\alpha_{k}$ if $i\in J_{k}$. Then, for brevity, we
denote $\beta_{i}=({\lambda_{i}^{\hat{{\bf B}}}-\lambda_{i}^{\tilde{{\bf
B}}}})/{\rho_{i}}$ and obtain that

$\displaystyle f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}})=\int_{0}^{\beta_{i}\rho_{i}}f_{l}^{\prime}(t+\lambda_{i}^{\tilde{{\bf
B}}})dt$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\beta_{i}\rho_{i}f_{l}^{\prime}(\beta_{i}\rho_{i}s+\lambda_{i}^{\tilde{{\bf
B}}})ds=\beta_{i}\rho_{i}f_{l}^{\prime}(\rho_{i})\int_{0}^{1}\frac{f_{l}^{\prime}(\rho_{i}(\beta_{i}s+\frac{\lambda_{i}^{\tilde{{\bf
B}}}}{\rho_{i}}))}{f_{l}^{\prime}(\rho_{i})}ds.$ (6.28)

Since $\beta_{i}=o_{p}(n^{-\frac{1}{2}})$, $\frac{\lambda_{i}^{\hat{{\bf
B}}}}{\rho_{i}}$ tends to 1, then from Assumption 4 we obtain

$\displaystyle\sqrt{n}\sum_{i=1}^{M}\dfrac{\left|f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}})\right|}{f_{l}^{\prime}(\rho_{i})\rho_{i}}=o_{p}(1).$ (6.29)

Then we have $\frac{\left|\sum_{i=1}^{M}(f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}}))\right|}{\varsigma_{l}}\leq\frac{\sum_{i=1}^{M}\left|f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}})\right|}{\varsigma_{l}}\leq\sum_{i=1}^{M}\frac{\left|f_{l}(\lambda_{i}^{\hat{{\bf
B}}})-f_{l}(\lambda_{i}^{\tilde{{\bf
B}}})\right|}{f_{l}^{\prime}(\rho_{i})\rho_{i}/\sqrt{n}}=o_{p}(1).$ Thus, it
is concluded that the procedure of truncation does not affect the limiting
distribution of LSS.

Therefore, in the following proofs, we can safely assume that
$\left|x_{ij}\right|<\eta_{n}\sqrt{n}$.

[Acknowledgments] The authors would like to thank Professor Jeff Yao for many
helpful suggestions and discussions. Zhidong Bai was partially supported by
NSFC Grants No.12171198, No.12271536, and Team Project of Jilin Provincial
Department of Science and Technology No.20210101147JC. Jiang Hu was partially
supported by NSFC Grant No.12171078.

Supplement to “A CLT for the LSS of large dimensional sample covariance
matrices with diverging spikes” This supplementary document contains proofs of
Theorems 3.1–4.5, and some useful lemmas. We also report some additional
simulation results in this document.

## References

  * Anderson (2003) Theodore Wilbur Anderson.  _An Introduction to Multivariate Statistical Analysis. Third Edition_.  Wiley New York, 2003.
  * Bai and Ng (2002) Jushan Bai and Serena Ng.  Determining the number of factors in approximate factor models.  _Econometrica_ , 70(1):191–221, 2002.
  * Bai (1999) Zhidong Bai  Methodologies in spectral analysis of large dimensional random matrices, A review.  _Statistica Sinica_ , 9:611–677, 1999.
  * Bai et al. (2007) Zhidong Bai, Baiqi Miao, and Guangming Pan.  On asymptotics of eigenvectors of large sample covariance matrix.  _The Annals of Probability_ , 35(4):1532–1572, 2007.
  * Bai et al. (2015) Zhidong Bai, Jiang Hu, Guangming Pan, and Wang Zhou.  Convergence of the empirical spectral distribution function of beta matrices.  _Bernoulli_ , 21(3):1538–1574, 2015.
  * Bai et al. (2019) Zhidong Bai, Huiqin Li, and Guangming Pan.  Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices.  _Bernoulli_ , 25(3):1838–1869, 2019.
  * Bai and Silverstein (2004) Zhidong Bai and Jack W. Silverstein.  CLT for linear spectral statistics of large-dimensional sample covariance matrices.  _The Annals of Probability_ , 32(1A):553 – 605, 2004.
  * Bai and Yao (2008) Zhidong Bai and Jianfeng Yao.  Central limit theorems for eigenvalues in a spiked population model.  _Annales de l’Institut Henri Poincaré, Probabilités et Statistiques_ , 44(3):447 – 474, 2008.
  * Bai and Yao (2012) Zhidong Bai and Jianfeng Yao.  On sample eigenvalues in a generalized spiked population model.  _Journal of Multivariate Analysis_ , 106:167–177, 2012\.
  * Bai et al. (2009) Zhidong Bai, Dandan Jiang, Jianfeng Yao, and Shurong Zheng.  Corrections to LRT on large-dimensional covariance matrix by RMT.  _The Annals of Statistics_ , 37(6B):3822 – 3840, 2009.
  * Baik and Silverstein (2006) Jinho Baik and Jack W. Silverstein.  Eigenvalues of large sample covariance matrices of spiked population models.  _Journal of Multivariate Analysis_ , 97(6):1382–1408, 2006.
  * Baik et al. (2018) Jinho Baik, Ji Oon Lee, and Hao Wu.  Ferromagnetic to paramagnetic transition in spherical spin glass.  _Journal of Statistical Physics_ , 173(5):1484–1522, 2018.
  * Banna et al. (2020) Marwa Banna, Jamal Najim, and Jianfeng Yao.  A clt for linear spectral statistics of large random information-plus-noise matrices.  _Stochastic Processes and their Applications_ , 130(4):2250–2281, 2020.
  * Bao et al. (2022) Zhigang Bao, Jiang Hu, Xiaocong Xu, and Xiaozhuo Zhang.  Spectral statistics of sample block correlation matrices.  arXiv:2207.06107, 2022.
  * Bloemendal et al. (2016) Alex Bloemendal, Antti Knowles, Horng-Tzer Yau, and Jun Yin.  On the principal components of sample covariance matrices.  _Probability Theory and Related Fields_ , 164(1-2):459–552, 2016.
  * Cai et al. (2020) T. Tony Cai, Xiao Han, and Guangming Pan.  Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices.  _The Annals of Statistics_ , 48(3):1255–1280, 2020.
  * Chen and Pan (2015) Binbin Chen and Guangming Pan.  Clt for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size.  _Bernoulli_ , 21(2):1089–1133, 2015.
  * Ding and Yang (2018) Xiucai Ding and Fan Yang  A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices.  _The Annals of Applied Probability_ , 28(3):1679–1738, 2018.
  * Dobriban (2020) Edgar Dobriban.  Permutation methods for factor analysis and pca.  _The Annals of Statistics_ , 48(5):2824–2847, 2020.
  * Donoho et al. (2018) David Donoho, Matan Gavish, and Iain Johnstone.  Optimal shrinkage of eigenvalues in the spiked covariance model.  _The Annals of Statistics_ , 46(4):1742–1778, 2018.
  * Gao et al. (2017) Jiti Gao, Xiao Han, Guangming Pan, and Yanrong Yang.  High dimensional correlation matrices: The central limit theorem and its applications.  _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 79(3):677–693, 2017.
  * Hu et al. (2019) Jiang Hu, Weiming Li, Zhi Liu, and Wang Zhou.  High-dimensional covariance matrices in elliptical distributions with application to spherical test.  _The Annals of Statistics_ , 47(1):527–555, 2019\.
  * Jiang and Bai (2021) Dandan Jiang and Zhidong Bai.  Generalized four moment theorem and an application to clt for spiked eigenvalues of high-dimensional covariance matrices.  _Bernoulli_ , 27(1):274–294, 2021\.
  * Jiang and Yang (2013) Tiefeng Jiang and Fan Yang.  Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions.  _The Annals of Statistics_ , 41(4):2029–2074, 2013.
  * Johnstone (2001) Iain M. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis.  _The Annals of Statistics_ , 29(2):295 – 327, 2001.
  * Johnstone (2008) Iain M. Johnstone.  Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence.  _The Annals of Statistics_ , 36(6):2638 – 2716, 2008.
  * Johnstone and Nadler (2017) Iain M. Johnstone and Boaz Nadler.  Roy’s largest root test under rank-one alternatives.  _Biometrika_ , 104(1):181–193, 2017.
  * Johnstone and Onatski (2020) Iain M. Johnstone and Alexei Onatski.  Testing in high-dimensional spiked models.  _The Annals of Statistics_ , 48(3):1231–1254, 2020\.
  * Johnstone and Paul (2018) Iain M. Johnstone and Debashis Paul.  PCA in high dimensions: An orientation.  _Proceedings of the IEEE_ , 106(8):1277–1292, 2018.
  * Jung and Marron (2009) Sungkyu Jung and J. S. Marron.  PCA consistency in high dimension, low sample size context.  _The Annals of Statistics_ , 37(6B):4104–4130, 2009.
  * Ledoit and Wolf (2002) Olivier Ledoit and Michael Wolf.  Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size.  _The Annals of Statistics_ , 30(4):1081–1102, 2002.
  * Li and Bai (2015) Huiqin Li and Zhidong Bai.  Extreme eigenvalues of large dimensional quaternion sample covariance matrices.  _Journal of Statistical Planning and Inference_ , 159:1–14, 2015.
  * Li et al. (2020) Zeng Li, Fang Han, and Jianfeng Yao.  Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model.  _The Annals of Statistics_ , 48(6):3138–3160, 2020.
  * Li et al. (2021) Zeng Li, Qinwen Wang, and Runze Li.  Central limit theorem for linear spectral statistics of large dimensional kendall’s rank correlation matrices and its applications.  _The Annals of Statistics_ , 49(3):1569–1593, 2021.
  * Nadler (2008) Boaz Nadler.  Finite sample approximation results for principal component analysis: A matrix perturbation approach.  _The Annals of Statistics_ , 36(6):2791–2817, 2008.
  * Nagao (1973) Hisao Nagao.  On Some Test Criteria for Covariance Matrix  _The Annals of Statistics_ 1(4):700–709, 1973.
  * Najim and Yao (2016) Jamal Najim and Jianfeng Yao.  Gaussian fluctuations for linear spectral statistics of large random covariance matrices.  _The Annals of Applied Probability_ , 26(3):1837–1887, 2016.
  * Olson (1974) Chester L. Olson.  Comparative robustness of six tests in multivariate analysis of variance  _Journal of the American Statistical Association_ , 69, 894–908, 1974.
  * Onatski et al. (2013) Alexei Onatski, Marcelo J. Moreira, and Marc Hallin.  Asymptotic power of sphericity tests for high-dimensional data.  _The Annals of Statistics_ , 41(3):1204–1231, 2013\.
  * Onatski et al. (2014) Alexei Onatski, Marcelo J. Moreira, and Marc Hallin.  Signal detection in high dimension: The multispiked case.  _The Annals of Statistics_ , 42(1):225–254, 2014\.
  * Pan (2014) Guangming Pan.  Comparison between two types of large sample covariance matrices.  _Annales de l’Institut Henri Poincaré, Probabilités et Statistiques_ , 50(2):655–677, 2014.
  * Pan and Zhou (2008) Guangming Pan and Wang Zhou.  Central limit theorem for signal-to-interference ratio of reduced rank linear receiver.  _The Annals of Applied Probability_ , 18(3):1232–1270, 2008.
  * Paul (2007) Debashis Paul.  Asymptotics of sample eigenstructure for a large dimensional spiked covariance model.  _Statistica Sinica_ , 17(4):1617–1642, 2007.
  * Perry et al. (2018) Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, and Ankur Moitra.  Optimality and sub-optimality of pca i: Spiked random matrix models.  _The Annals of Statistics_ , 46(5):2416–2451, 2018.
  * Silverstein (1995) Jack W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices.  _Journal of Multivariate Analysis_ , 55(2):331–339, 1995.
  * Wang and Yao (2013) Qinwen Wang and Jianfeng Yao.  On the sphericity test with large-dimensional observations.  _Electronic Journal of Statistics_ , 7:2164–2192, 2013\.
  * Wang and Yao (2017) Qinwen Wang and Jianfeng Yao.  Extreme eigenvalues of large-dimensional spiked fisher matrices with application.  _The Annals of Statistics_ , 45(1):415–460, 2017\.
  * Wang and Fan (2017) Weichen Wang and Jianqing Fan.  Asymptotics of empirical eigenstructure for high dimensional spiked covariance.  _The Annals of Statistics_ , 45(3):1342–1374, 2017\.
  * Wilks (1938) Samuel S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses.  _The Annals of Mathematical Statistics_ , 9(1):60–62, 1938.
  * Yang and Johnstone (2018) Jeha Yang and Iain M. Johnstone.  Edgeworth correction for the largest eigenvalue in a spiked pca model.  _Statistica Sinica_ , 2018.
  * Yang and Pan (2015) Yanrong Yang and Guangming Pan.  Independence test for high dimensional data based on regularized canonical correlation coefficients.  _The Annals of Statistics_ , 43(2):467–500, 2015\.
  * Yao et al. (2015) Jianfeng Yao, Shurong Zheng, and Zhidong Bai.  _Large Sample Covariance Matrices and High-Dimensional Data Analysis_.  Cambridge University Press, 2015.
  * Yao et al. (2018) Zhigang Yao, Ye Zhang, Zhidong Bai, and William F. Eddy.  Estimating the number of sources in magnetoencephalography using spiked population eigenvalues.  _Journal of the American Statistical Association_ , 113(522):505–518, 2018.
  * Yin (2021) Yanqing Yin.  Spectral statistics of high dimensional sample covariance matrix with unbounded population spectral norm.  _Bernoulli_ , 28(3):1729–1756, 2022.
  * Zhang et al. (2022) Zhixiang Zhang, Shurong Zheng, Guangming Pan, and Pingshou Zhong.  Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices.  _The Annals of Statistics_ , 50(4):2205–2230, 2022.
  * Zheng (2012) Shurong Zheng.  Central limit theorems for linear spectral statistics of large dimensional F-matrices.  _Annales de l’Institut Henri Poincaré, Probabilités et Statistiques_ , 48(2):444–476, 2012.
  * Zheng et al. (2015) Shurong Zheng, Zhidong Bai, and Jianfeng Yao.  Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing.  _The Annals of Statistics_ , 43(2):546 – 591, 2015.
  * Zheng et al. (2019) Shurong Zheng, Guanghui Cheng, Jianhua Guo, and Hongtu Zhu.  Test for high-dimensional correlation matrices.  _The Annals of Statistics_ , 47(5):2887–2921, 2019.
  * Zhou and Marron (2015) Yihui Zhou and James Stephen Marron.  High dimension low sample size asymptotics of robust PCA.  _Electronic Journal of Statistics_ , 9(1):204–218, 2015\.

, , , and

In this document we present some technical details involved in Liu et al.
(2022). More precisely, in Section 7, we prove Theorems 3.1–4.5 of the main
file. Some derivations and calculations in Section 7 are postponed to Section
8. In Section 9 we provide some useful lemmas. Finally, in Section 10, we
report some additional simulation results in this part.

The number of scheme(equations,theorems,lemmas,etc.) is shared with the main
document so that there are no misunderstandings with the use of references.

##  7 Proofs of Theorems 3.1–4.5

###  7.1 Proof of Theorem 3.1

The proof of Theorem 3.1 builds on the decomposition analysis of the LSSs and
it is divided into part (I) $\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)$ and
part (II) $\sum_{j=M+1}^{p}f\left({\bf\lambda}_{j}\right)$. Enlightened by the
BST in Bai and Silverstein (2004), we have

$\displaystyle\sum_{j=1}^{p}f\left({\bf\lambda}_{j}\right)-p\int
f(x)dF^{c_{n},H_{n}}$
$\displaystyle=\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)+\sum_{j=M+1}^{p}f\left({\bf\lambda}_{j}\right)-p\int
f(x)dF^{c_{n},H_{n}}$
$\displaystyle=\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)+\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f(x)dF^{c_{nM},H_{2n}}+\sum_{j=M+1}^{p}f\left({\bf\lambda}_{j}\right)-\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)$
$\displaystyle+(p-M)\int f(x)dF^{c_{nM},H_{2n}}-p\int f(x)dF^{c_{n},H_{n}}.$

Since Lemma 6.1 has shown the difference between $(p-M)\int
f(x)dF^{c_{nM},H_{2n}}$ and $p\int f(x)dF^{c_{n},H_{n}}$ is 0. Moreover, in
Lemma 6.2 we have proved

$\sum_{j=M+1}^{p}f\left({\bf\lambda}_{j}\right)-\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)=\frac{M}{2\pi
i}\oint_{C}f\left(z\right)\frac{\underline{m}_{2n0}^{{}^{\prime}}(z)}{\underline{m}_{2n0}(z)}dz+o_{p}(1).$

It follows that

$\displaystyle\sum_{j=1}^{p}f\left({\bf\lambda}_{j}\right)-p\int
f(x)dF^{c_{n},H_{n}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)+\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f(x)dF^{c_{nM},H_{2n}}+\frac{M}{2\pi
i}\oint_{C}f\left(z\right)\frac{\underline{m}_{2n0}^{{}^{\prime}}(z)}{\underline{m}_{2n0}(z)}dz+o_{p}(1),$

which yields

$\displaystyle\sum_{j=1}^{p}f\left({\bf\lambda}_{j}\right)-p\int
f(x)dF^{c_{n},H_{n}}-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left(\alpha_{k}\right)\right)-\frac{M}{2\pi
i}\oint_{C}f\left(z\right)\frac{\underline{m}_{2n0}^{{}^{\prime}}(z)}{\underline{m}_{2n0}(z)}dz$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left(\alpha_{k}\right)\right)+\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f(x)dF^{c_{nM},H_{2n}}+o_{p}(1).$ (7.30)

The analysis below is executed by dividing (7.30) into two parts: (I)
$\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left(\alpha_{k}\right)\right)$
and (II) $\sum_{j=1}^{p-M}f\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f(x)dF^{c_{nM},H_{2n}}$, where we ignore the impact of $o_{p}(1)$ on the
asymptotic distribution. Since we have derived the asymptotic distribution of
part (II) in Lemma 6.5, we only need to consider the asymptotic distribution
of part (I)
$\sum_{j=1}^{M}f\left(\lambda_{j}\right)-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)$.
From the proof of Lemma 6.4,
$\sum_{j=1}^{M}f\left({\bf\lambda}_{j}\right)-\sum_{k=1}^{K}d_{k}f\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)$
has the same limiting distribution as $\sum_{k=1}^{K}\varpi_{nk}\sum_{j\in
J_{k}}\gamma_{kj}$. From Lemma 6.3, we have $\left(\gamma_{kj},j\in
J_{k}\right)^{\top}$ converges weakly to the joint distribution of the $d_{k}$
eigenvalues of Gaussian random matrix
$-\frac{1}{\theta_{k}}\left[{\bf\Omega}_{\phi_{k}}\right]_{kk}$, so
$\sum_{j\in J_{k}}\gamma_{kj}\stackrel{{\scriptstyle
d}}{{\rightarrow}}-\frac{1}{\theta_{k}}\mathrm{tr}\left[{\bf\Omega}_{\phi_{k}}\right]_{kk}$.
Recall that $\omega_{ij}$ is the element of ${\bf\Omega}_{\phi_{k}}$, and
$\mathrm{tr}\left[{\bf\Omega}_{\phi_{k}}\right]_{kk}$ is the summation of the
diagonal element, that is, $\sum_{j\in J_{k}}\omega_{jj}$. Since the diagonal
elements are i.i.d., then $\mathbb{E}\left(\sum_{j\in
J_{k}}\omega_{jj}\right)=0$, $\mathrm{Var}\left(\sum_{j\in
J_{k}}\omega_{jj}\right)=\sum_{j\in
J_{k}}\mathrm{Var}\left(\omega_{jj}\right)+\sum_{j_{1}\neq
j_{2}}\mathrm{cov}\left(\omega_{j_{1}j_{1}},\omega_{j_{2}j_{2}}\right)=\sum_{j\in
J_{k}}(\left(\alpha_{x}+1\right)\theta_{k}+\beta_{x}\mathcal{U}_{jjjj}\nu_{k})+\sum_{j_{1}\neq
j_{2}}\beta_{x}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}\nu_{k}=\left(\alpha_{x}+1\right)\theta_{k}d_{k}+\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}\beta_{x}\nu_{k}$

Therefore, from Lemma 6.3, we have that the asymptotic distribution of
$\sum_{j\in J_{k}}\gamma_{kj}$ is a Gaussian distribution with

$\mathbb{E}\sum_{j\in J_{k}}\gamma_{kj}=0,$
$s_{k}^{2}\triangleq\mathrm{Var}\left(\sum_{j\in
J_{k}}\gamma_{kj}\right)=\frac{\left(\alpha_{x}+1\right)\theta_{k}d_{k}+\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}\beta_{x}\nu_{k}}{\theta_{k}^{2}},$

and then, we directly derive that the mean function of
$\sum_{k=1}^{K}\varpi_{nk}\sum_{j\in J_{k}}\gamma_{kj}$ is 0 and that its
covariance function is

$\mathrm{Var}\left(Y_{f_{1}}\right)=\sum_{k=1}^{K}\varpi_{nk1}^{2}s_{k}^{2}.$

Finally, we focus on the asymptotic distribution of equation (7.30). Because
of Lemma 6.4, the two LSSs are asymptotically independent; thus, the random
variable

$\varsigma_{1}^{-1}\left(Y_{1}-\mathbb{E}Y_{1}\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N\left(0,1\right)$

with mean function $\mathbb{E}Y_{1}=\mu_{1}$ being

$\displaystyle-\frac{\alpha_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}f_{1}(z)\frac{c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{\left(1-c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)\left(1-\alpha_{x}c_{nM}\int\frac{\underline{m}_{2n0}^{2}(z)t^{2}}{\left(1+t\underline{m}_{2n0}(z)\right)^{2}}dH_{2n}(t)\right)}dz$
$\displaystyle-\frac{\beta_{x}}{2\pi
i}\cdot\oint_{\mathcal{C}}f_{1}(z)\frac{c_{nM}\int\underline{m}_{2n0}^{3}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-3}dH_{2n}(t)}{1-c_{nM}\int\underline{m}_{2n0}^{2}(z)t^{2}\left(1+t\underline{m}_{2n0}(z)\right)^{-2}dH_{2n}(t)}dz,$

and covariance function $\varsigma_{1}^{2}$ being

$\displaystyle\sum_{k=1}^{K}\varpi_{nk1}^{2}s_{k}^{2}-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{1}\left(z_{1}\right)f_{1}\left(z_{2}\right)\vartheta_{n}^{2}dz_{1}dz_{2},$

where $\vartheta_{n}^{2}$ is defined in Lemma 6.5. Therefore, the proof is
finished.

###  7.2 Proof of Theorem 3.2

Similar to the proof of Theorem 3.1, we divide the LSSs into two parts.
Different from the above analysis, in this section, we focus on the
multidimensional case under Assumptions 1–6. Recall that we defined

$G_{n}\left(x\right)=p\left[F^{{\bf
B}}\left(x\right)-F^{c_{n},H_{n}}\left(x\right)\right],$ $Y_{l}=\int
f_{l}\left(x\right)dG_{n}\left(x\right)-\sum_{k=1}^{K}d_{k}f_{l}\left(\phi_{n}\left(\alpha_{k}\right)\right)-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{l}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz.$

Because of equation (7.30), the random vector $\left(Y_{1},\dots,Y_{h}\right)$
shares the same asymptotic distribution with the summation of two random
vectors

$(\sum_{k=1}^{K}\varpi_{nk1}\sum_{j\in
J_{k}}\gamma_{kj},\dots,\sum_{k=1}^{K}\varpi_{nkh}\sum_{j\in
J_{k}}\gamma_{kj})$

and

$(\sum_{j=1}^{p-M}f_{1}\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f_{1}(x)dF^{c_{nM},H_{2n}},\dots,\sum_{j=1}^{p-M}f_{h}\left(\widetilde{{\bf\lambda}}_{j}\right)-(p-M)\int
f_{h}(x)dF^{c_{nM},H_{2n}}).$

First, we focus on the first random vector. Similar to the proof of Theorem
3.1, we derive that the mean function of the first random vector is 0 and that
the covariance function is

$\mathrm{Cov}\left(Y_{f_{s}},Y_{f_{t}}\right)=\sum_{k=1}^{K}\frac{\phi_{n}^{2}\left({\bf\alpha}_{k}\right)}{n}f_{s}^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)f_{t}^{\prime}\left(\phi_{n}\left({\bf\alpha}_{k}\right)\right)s_{k}^{2}=\sum_{k=1}^{K}\varpi_{nks}\varpi_{nkt}s_{k}^{2},$

Moreover, the asymptotic distribution of the second random vector is derived
in Zheng et al. (2015). Because of Lemma 6.4, two random vectors are
asymptotically independent; thus, the random vector

$\left(Y_{1}-\mathbb{E}Y_{1},\dots,Y_{h}-\mathbb{E}Y_{h}\right)^{\top}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{h}\left(0,{\bf\Omega}\right),$

with mean function $\mathbb{E}Y_{l}$ is the same as $\mu_{l}$, and the
covariance matrix is ${\bf\Omega}$ with its entries

$\displaystyle\omega_{st}=\sum_{k=1}^{K}\varpi_{nks}\varpi_{nkt}s_{k}^{2}-\kappa_{nst},$

where

$\displaystyle\kappa_{nst}$
$\displaystyle=\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\frac{f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)}{\left(\underline{m}_{2n0}\left(z_{1}\right)-\underline{m}_{2n0}\left(z_{2}\right)\right)^{2}}d\underline{m}_{2n0}\left(z_{1}\right)d\underline{m}_{2n0}\left(z_{2}\right)+\frac{c_{nM}\beta_{x}}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{s}\left(z_{1}\right)$
$\displaystyle
f_{t}\left(z_{2}\right)\left[\int\frac{t}{\left(\underline{m}_{2n0}\left(z_{1}\right)t+1\right)^{2}}\right.\left.\times\frac{t}{\left(\underline{m}_{2n0}\left(z_{2}\right)t+1\right)^{2}}dH_{2n}(t)\right]d\underline{m}_{2n0}\left(z_{1}\right)d\underline{m}_{2n0}\left(z_{2}\right)$
$\displaystyle\quad+\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}f_{s}\left(z_{1}\right)f_{t}\left(z_{2}\right)\left[\frac{\partial^{2}}{\partial
z_{1}\partial
z_{2}}\log\left(1-a_{n}\left(z_{1},z_{2}\right)\right)\right]dz_{1}dz_{2},$
$\displaystyle a_{n}\left(z_{1},z_{2}\right)$
$\displaystyle=\alpha_{x}\left(1+\frac{\underline{m}_{2n0}\left(z_{1}\right)\underline{m}_{2n0}\left(z_{2}\right)\left(z_{1}-z_{2}\right)}{\underline{m}_{2n0}\left(z_{2}\right)-\underline{m}_{2n0}\left(z_{1}\right)}\right).$

Then, we obtain the random vector

$\left(\frac{Y_{1}-\mathbb{E}Y_{1}}{\sigma_{1}},\dots,\frac{Y_{h}-\mathbb{E}Y_{h}}{\sigma_{h}}\right)^{\top}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{h}\left(0,{\bf\Psi}\right),$

which has a mean function that is the same as that in Theorem 3.1, and
variance function

$\displaystyle\sigma_{l}^{2}$
$\displaystyle=\sum_{k=1}^{K}\varpi_{nkl}^{2}s_{k}^{2}-\kappa_{nll},\quad
l=1,\dots,h,$

and the covariance matrix ${\bf\Psi}=\left(\psi_{st}\right)_{h\times h}$ is
the correlation coefficient matrix of random vector
$(Y_{1},\dots,Y_{h})^{\top}$ with its entries
$\psi_{st}=\lim_{n\rightarrow\infty}\psi_{nst},$

$\displaystyle\psi_{nst}=\frac{\sum_{k=1}^{K}\varpi_{nks}\varpi_{nkt}s_{k}^{2}-\kappa_{nst}}{\sqrt{\sum_{k=1}^{K}\varpi_{nks}^{2}s_{k}^{2}-\kappa_{nss}}\sqrt{\sum_{k=1}^{K}\varpi_{nkt}^{2}s_{k}^{2}-\kappa_{ntt}}},$

Note that renormalization is necessary to guarantee that elements in the
correlation coefficient matrix ${\bf\Psi}$ are limited. Therefore, the proof
is finished.

###  7.3 Proof of Theorem 4.1

The result under $H_{0}$ is a direct result of Theorem 4.1 in Zheng et al.
(2015) using the substitution principle. Therefore, we omit the proof here.
Next, we focus on the result under $H_{1}.$ Recall that

$G_{n}\left(x\right)=p\left[F^{{\bf
B}}\left(x\right)-F^{c_{n},H_{n}}\left(x\right)\right],$ $Y=\int
f_{L}\left(x\right)dG_{n}\left(x\right)-\sum_{k=1}^{K}d_{k}f_{L}\left(\phi_{n}\left(\alpha_{k}\right)\right)-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{L}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz,$

when $f_{L}(x)=x-\log x-1.$ After some calculations, we obtain

$\displaystyle\int f_{L}\left(x\right)dG_{n}\left(x\right)=\mathrm{tr}{\bf
B}-\log\left|{\bf B}\right|-p-p\int f_{L}(x)dF^{c_{n},H_{n}}(x)=L-p\int
f_{L}(x)dF^{c_{n},H_{n}}(x),$ $\displaystyle p\int
f_{L}(x)dF^{c_{n},H_{n}}(x)=(p-M)(1-\frac{c_{nM}-1}{c_{nM}}\log\left(1-c_{nM}\right)),$
(7.31)
$\displaystyle\sum_{k=1}^{K}d_{k}f_{L}\left(\phi_{n}\left(\alpha_{k}\right)\right)=\sum_{k=1}^{K}d_{k}\left(\phi_{n}\left(\alpha_{k}\right)-\log\phi_{n}\left(\alpha_{k}\right)-1\right),$
$\displaystyle\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{L}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz=-M(c_{nM}+\log(1-c_{nM})),$
(7.32)

where (7.31) is obtained from Lemma 6.1 and Bai et al. (2009). For
consistency, we present the proof of (7.32) in Section 8. According to Theorem
3.1, since $f_{L}(x)=x-\log x-1$, ${\bf D}_{2}={\bf I}_{p-M},$
${\bf\Gamma}={\bf V}_{2}{\bf U}_{2}^{*}$, then we have

$\displaystyle\frac{L-p\int
f_{L}(x)dF^{c_{n},H_{n}}(x)-\breve{\mu}_{l}}{\breve{\varsigma}_{l}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(0,1),$

where the mean function is
$\breve{\mu}_{l}=-\frac{\log\left(1-c_{nM}\right)}{2}\alpha_{x}+\frac{c_{nM}}{2}\beta_{x}+\sum_{k=1}^{K}d_{k}(\phi_{n}\left({\alpha}_{k}\right)-\log\phi_{n}\left({\alpha}_{k}\right)-1)-M(c_{nM}+\log(1-c_{nM}))$.
For covariance term, $\breve{\varsigma}_{l}^{2}$ equals
$\sum_{k=1}^{K}\frac{\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\left(z_{1}-\log\left(z_{1}\right)-1\right)\left(z_{2}-\log\left(z_{2}\right)-1\right)\vartheta_{n}^{2}dz_{1}dz_{2},$
where
$s_{k}^{2}=\frac{\left(\alpha_{x}+1\right)d_{k}}{\theta_{k}}+\frac{\sum_{j_{1},j_{2}\in
J_{k}}\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}\beta_{x}\nu_{k}}{\theta_{k}^{2}}$,
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}),$
and

$\displaystyle\Theta_{0,n}(z_{1},z_{2})=$
$\displaystyle\dfrac{\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{(\underline{m}_{2n0}(z_{1})-\underline{m}_{2n0}(z_{2}))^{2}}-\dfrac{1}{(z_{1}-z_{2})^{2}}.$

For $\Theta_{1,n}(z_{1},z_{2}),$ since

$\displaystyle{\bf P}_{n}\left(z\right)=$
$\displaystyle\left(\left(1-c_{nM}\right){\bf V}_{2}{\bf
V}_{2}^{*}-zc_{nM}m_{2n0}\left(z\right){\bf V}_{2}{\bf V}_{2}^{*}-z{\bf
I}_{p}\right)^{-1},$

then

$\displaystyle{\mathcal{A}_{n}(z_{1},z_{2})}=$
$\displaystyle\frac{z_{1}z_{2}}{n}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\mathrm{tr}{\bf
U}_{2}{\bf V}_{2}^{*}{\bf P}_{n}\left(z_{1}\right){\bf V}_{2}{\bf
U}_{2}^{*}\bar{{\bf U}}_{2}{\bf V}_{2}^{\top}{\bf
P}_{n}^{\top}\left(z_{2}\right)\bar{{\bf V}}_{2}{\bf U}_{2}^{\top},$
$\displaystyle=$
$\displaystyle\frac{z_{1}z_{2}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\mathrm{tr}{\bf
U}_{2}{\bf U}_{2}^{*}\bar{{\bf U}}_{2}{\bf
U}_{2}^{\top}}{n\left(1-c_{nM}-z_{1}c_{nM}m_{2n0}(z_{1})-z_{1}\right)\left(1-c_{nM}-z_{2}c_{nM}m_{2n0}(z_{2})-z_{2}\right)},$
$\displaystyle=$
$\displaystyle\frac{\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})}{n\left(1+\underline{m}_{2n0}(z_{1})\right)\left(1+\underline{m}_{2n0}(z_{2})\right)}\mathrm{tr}{\bf
U}_{2}{\bf U}_{2}^{*}\bar{{\bf U}}_{2}{\bf U}_{2}^{\top}.$

For $\mathrm{tr}{\bf U}_{2}{\bf U}_{2}^{*}\bar{{\bf U}}_{2}{\bf
U}_{2}^{\top},$ since ${\bf U}{\bf U}^{*}={\bf I}_{p},$ therefore
$\mathrm{tr}{\bf U}_{2}{\bf U}_{2}^{*}\bar{{\bf U}}_{2}{\bf
U}_{2}^{\top}=\mathrm{tr}\left({\bf I}_{p}-{\bf U}_{1}{\bf
U}_{1}^{*}\right)\left({\bf I}_{p}-{\bf U}_{1}{\bf
U}_{1}^{*}\right)^{\top}=p-\mathrm{tr}\left({\bf U}_{1}{\bf
U}_{1}^{*}\right)^{\top}-\mathrm{tr}{\bf U}_{1}{\bf U}_{1}^{*}+\mathrm{tr}{\bf
U}_{1}{\bf U}_{1}^{*}\left({\bf U}_{1}{\bf U}_{1}^{*}\right)^{\top}.$
Moreover, since $\mathrm{tr}{\bf U}_{1}{\bf U}_{1}^{*}=M,$ $\mathrm{tr}{\bf
U}_{1}{\bf U}_{1}^{*}\left({\bf U}_{1}{\bf U}_{1}^{*}\right)^{\top}\\\
=\sum_{s,t=1}^{p}\left(\sum_{i=1}^{M}u_{si}\bar{u}_{ti}\right)^{2}.$
Therefore,
${\mathcal{A}}_{n\left(z_{1},z_{2}\right)}=\frac{p-2M+\sum_{s,t=1}^{p}\left(\sum_{i=1}^{M}u_{si}\bar{u}_{ti}\right)^{2}}{n}\frac{\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})}{\left(1+\underline{m}_{2n0}(z_{1})\right)\left(1+\underline{m}_{2n0}(z_{2})\right)}.$
Denote
$\tilde{c}=\frac{p-2M+\sum_{s,t=1}^{p}\left(\sum_{i=1}^{M}u_{si}\bar{u}_{ti}\right)^{2}}{n},$
then
${\mathcal{A}}_{n}\left(z_{1},z_{2}\right)=\frac{\tilde{c}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})}{\left(1+\underline{m}_{2n0}(z_{1})\right)\left(1+\underline{m}_{2n0}(z_{2})\right)}.$
Therefore

$\displaystyle\Theta_{1,n}(z_{1},z_{2})=$
$\displaystyle\frac{\partial}{\partial
z_{2}}\left\\{\dfrac{\partial\mathcal{A}_{n}(z_{1},z_{2})}{\partial
z_{1}}\dfrac{1}{1-\alpha_{x}\mathcal{A}_{n}(z_{1},z_{2})}\right\\},$
$\displaystyle=$
$\displaystyle\dfrac{\tilde{c}\underline{m}^{\prime}_{2n0}(z_{1})\underline{m}^{\prime}_{2n0}(z_{2})}{\left(\left(1+\underline{m}_{2n0}(z_{1})\right)\left(1+\underline{m}_{2n0}(z_{2})\right)-\alpha_{x}\tilde{c}\underline{m}_{2n0}(z_{1})\underline{m}_{2n0}(z_{2})\right)^{2}}.$

For $\Theta_{2,n}(z_{1},z_{2}),$ since
$\Theta_{2,n}(z_{1},z_{2})=\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii},$ and

$\displaystyle{\bf\Gamma}^{*}{\bf P}_{n}^{2}(z_{1}){\bf\Gamma}={\bf U}_{2}{\bf
V}_{2}^{*}\left((1-c_{nM}){\bf V}_{2}{\bf
V}_{2}^{*}-z_{1}c_{nM}m_{2n0}(z_{1}){\bf V}_{2}{\bf V}_{2}^{*}-z{\bf
I}_{p}\right)^{-2}{\bf V}_{2}{\bf U}_{2}^{*},$

by using lemma 9.2, we have

$\displaystyle{\bf
P}_{n}\left(z_{1}\right)=\dfrac{\underline{m}_{2n0}(z_{1})}{z\left(1+\underline{m}_{2n0}(z_{1})\right)}{\bf
V}_{2}{\bf V}_{2}^{*}-\frac{1}{z_{1}}{\bf I}_{p},$

then

$\displaystyle{\bf\Gamma}^{*}{\bf P}_{n}^{2}(z_{1}){\bf\Gamma}=$
$\displaystyle{\bf U}_{2}{\bf
V}_{2}^{*}(\dfrac{\underline{m}_{2n0}(z_{1})}{z_{1}\left(1+\underline{m}_{2n0}(z_{1})\right)}{\bf
V}_{2}{\bf V}_{2}^{*}-\frac{1}{z_{1}}{\bf
I}_{p})(\dfrac{\underline{m}_{2n0}(z_{1})}{z_{1}\left(1+\underline{m}_{2n0}(z_{1})\right)}{\bf
V}_{2}{\bf V}_{2}^{*}-\frac{1}{z_{1}}{\bf I}_{p}){\bf V}_{2}{\bf U}_{2}^{*},$
$\displaystyle=$ $\displaystyle{\bf U}_{2}{\bf
V}_{2}^{*}(\dfrac{\underline{m}^{2}_{2n0}(z_{1})}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
V}_{2}{\bf
V}_{2}^{*}-\dfrac{2\underline{m}_{2n0}(z_{1})}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)}{\bf
V}_{2}{\bf V}_{2}^{*}+\frac{1}{z_{1}^{2}}{\bf I}_{p}){\bf V}_{2}{\bf
U}_{2}^{*},$ $\displaystyle=$ $\displaystyle{\bf
U}_{2}(\dfrac{\underline{m}^{2}_{2n0}(z_{1})}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
I}_{p-M}-\dfrac{2\underline{m}_{2n0}(z_{1})}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)}{\bf
I}_{p-M}+\frac{1}{z_{1}^{2}}{\bf I}_{p-M}){\bf U}_{2}^{*},$ $\displaystyle=$
$\displaystyle\frac{1}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
U}_{2}{\bf U}_{2}^{*}.$

Therefore,

$\displaystyle\sum_{i=1}^{p}[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}]_{ii}[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}]_{ii}=\sum_{i=1}^{p}[\frac{1}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
U}_{2}{\bf
U}_{2}^{*}]_{ii}[\frac{1}{z_{2}^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}}{\bf
U}_{2}{\bf U}_{2}^{*}]_{ii}.$

Since

$\displaystyle[\frac{1}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
U}_{2}{\bf
U}_{2}^{*}]_{ii}=[\frac{1}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
I}_{p}-\frac{1}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}}{\bf
U}_{1}{\bf U}_{1}^{*}]_{ii}$
$\displaystyle=\dfrac{1-\sum_{j=1}^{M}\left|u_{ij}\right|^{2}}{z_{1}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}},$

then

$\displaystyle\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii}=\sum_{i=1}^{p}\dfrac{\left(1-\sum_{j=1}^{M}\left|u_{ij}\right|^{2}\right)^{2}}{z_{1}^{2}z_{2}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}},$
$\displaystyle=$
$\displaystyle\dfrac{p-2M+\sum_{i=1}^{p}\left(\sum_{j=1}^{M}\left|u_{ij}\right|\right)^{2}}{z_{1}^{2}z_{2}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}}=\dfrac{p-2M+\sum_{j_{1},j_{2}=1}^{M}{\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}}{z_{1}^{2}z_{2}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}}.$

Then

$\displaystyle\Theta_{2,n}(z_{1},z_{2})=$
$\displaystyle\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\sum_{i=1}^{p}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{1}){\bf\Gamma}\right]_{ii}\left[{\bf\Gamma}^{*}{\bf
P}_{n}^{2}(z_{2}){\bf\Gamma}\right]_{ii},$ $\displaystyle=$
$\displaystyle\dfrac{z_{1}^{2}z_{2}^{2}\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{n}\dfrac{p-2M+\sum_{j_{1},j_{2}=1}^{M}{\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}}{z_{1}^{2}z_{2}^{2}\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}}$
$\displaystyle=$
$\displaystyle\dfrac{p-2M+\sum_{j_{1},j_{2}=1}^{M}{\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}}{n}\dfrac{\underline{m}_{2n0}^{\prime}(z_{1})\underline{m}_{2n0}^{\prime}(z_{2})}{\left(1+\underline{m}_{2n0}(z_{1})\right)^{2}\left(1+\underline{m}_{2n0}(z_{2})\right)^{2}}.$

Since the covariance of bulk part is
$-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\left(z_{1}-\log\left(z_{1}\right)-1\right)\left(z_{2}-\log\left(z_{2}\right)-1\right)\vartheta_{n}^{2}dz_{1}dz_{2}$,
where
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}).$
By contour integral calculations, we obtain the covariance equals
$-\log(1-c_{nM})-c_{nM}+\alpha_{x}\left(-\log(1-\tilde{c})-\tilde{c}\right),$
where
$\tilde{c}=\dfrac{p-2M+\sum_{s,t=1}^{p}\left(\sum_{i=1}^{M}u_{si}\bar{u}_{ti}\right)^{2}}{n}.$
Since $\tilde{c}-c_{nM}\rightarrow 0$ as $n\rightarrow\infty,$ therefore

$\displaystyle\breve{\varsigma}_{l}^{2}=$
$\displaystyle\sum_{k=1}^{K}\frac{\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(-\log(1-c_{nM})-c_{nM}\right),$

and

$\displaystyle\frac{L-p\int
f_{L}(x)dF^{c_{n},H_{n}}(x)-\breve{\mu}_{l}}{\breve{\varsigma}_{l}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(0,1).$

The proof of Theorem 4.1 is finished.

###  7.4 Proof of Theorem 4.2

First, we focus on the results under $H_{0}$. From Lemma 9.1, we have

$\displaystyle I_{1}(f_{W})$ $\displaystyle=c,$ $\displaystyle I_{2}(f_{W})$
$\displaystyle=c,$ $\displaystyle J_{1}(f_{W},f_{W})$
$\displaystyle=4c^{3}+2c^{2},$ $\displaystyle J_{2}(f_{W},f_{W})$
$\displaystyle=4c^{3},$

which then yields

$\displaystyle\mu_{w}$
$\displaystyle=\alpha_{x}I_{1}(f_{W})+\beta_{x}I_{2}(f_{W})=\alpha_{x}c+\beta_{x}c,$
$\displaystyle\varsigma_{w}^{2}$
$\displaystyle=(\alpha_{x}+1)J_{1}(f_{W},f_{W})+\beta_{x}J_{2}(f_{W},f_{W})=(\alpha_{x}+1)(4c^{3}+2c^{2})+4\beta_{x}c^{3}.$

The results are still valid if $c$ is replaced by $c_{n}$. Moreover, the
center term

$\displaystyle\int f_{W}(x)dF^{c_{n},H_{n}}=c_{n},$ (7.33)

is a direct result of Lemma 2.2 in Wang and Yao (2013). Therefore, from Zheng
et al. (2015) or Wang and Yao (2013), we have

$\displaystyle\frac{W-p\int
f_{W}(x)dF^{c_{n},H_{n}}-\mu_{w}}{\varsigma_{w}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1).$

Then, we focus on the results under $H_{1}$. Note that

$\displaystyle Y=\int
f_{W}\left(x\right)dG_{n}\left(x\right)-\sum_{k=1}^{K}d_{k}f_{W}\left(\phi_{n}\left(\alpha_{k}\right)\right)-\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{W}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz.$

After some calculations, we obtain

$\displaystyle\int f_{W}\left(x\right)dG_{n}\left(x\right)=\mathrm{tr}({\bf
B}-{\bf I}_{p})^{2}-p\int f_{W}(x)dF^{c_{n},H_{n}}=W-p\int
f_{W}(x)dF^{c_{n},H_{n}},$ $\displaystyle p\int
f_{W}(x)dF^{c_{n},H_{n}}=(p-M)\int f_{W}(x)dF^{c_{nM},H_{2n}}=(p-M)c_{nM},$
$\displaystyle\sum_{k=1}^{K}d_{k}f_{W}\left(\phi_{n}\left(\alpha_{k}\right)\right)=\sum_{k=1}^{K}d_{k}\left(\phi_{n}^{2}\left(\alpha_{k}\right)-2\phi_{n}\left(\alpha_{k}\right)+1\right),$
$\displaystyle\frac{M}{2\pi
i}\oint_{\mathcal{C}}f_{W}\left(z\right)\frac{\underline{m}_{2n0}^{\prime}(z)}{\underline{m}_{2n0}(z)}dz=-Mc_{nM}^{2}.$
(7.34)

For consistency, we present the proof of (7.34) in Section 8. Therefore, from
Theorem 3.1, we have

$\displaystyle\dfrac{W-(p-M)\breve{\ell}_{w}-\breve{\mu}_{w}}{\breve{\varsigma}_{w}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

where

$\displaystyle\breve{\ell}_{w}$
$\displaystyle=c_{nM},~{}~{}~{}\breve{\mu}_{w}=\alpha_{x}c_{nM}+\beta_{x}c_{nM}+\sum_{k=1}^{K}d_{k}\left(\phi_{n}^{2}\left({\alpha}_{k}\right)-2\phi_{n}\left({\alpha}_{k}\right)+1\right)-Mc_{nM}^{2},$
$\displaystyle\breve{\varsigma}_{w}^{2}=$
$\displaystyle-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\left(z_{1}-1\right)^{2}\left(z_{2}-1\right)^{2}\vartheta_{n}^{2}dz_{1}dz_{2}+\sum_{k=1}^{K}\frac{4\phi_{n}^{2}\left(\alpha_{k}\right)\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}.$

Since
$\vartheta_{n}^{2}=\Theta_{0,n}(z_{1},z_{2})+\alpha_{x}\Theta_{1,n}(z_{1},z_{2})+\beta_{x}\Theta_{2,n}(z_{1},z_{2}),$
and $\Theta_{0,n}(z_{1},z_{2}),\Theta_{1,n}(z_{1},z_{2}),$ and
$\Theta_{2,n}(z_{1},z_{2})$ are calculated in the proof of Theorem 4.1, then
by some calculations we obtain
$-\frac{1}{4\pi^{2}}\oint_{\mathcal{C}_{1}}\oint_{\mathcal{C}_{2}}\left(z_{1}-1\right)^{2}\left(z_{2}-1\right)^{2}\vartheta_{n}^{2}dz_{1}dz_{2}$
equals
$4c_{nM}^{3}+2c_{nM}^{2}+\alpha_{x}\left(4\tilde{c}^{3}+2\tilde{c}^{2}\right)+4\beta_{x}\check{c}^{3},$
where
$\check{c}=\dfrac{p-2M+\sum_{j_{1},j_{2}=1}^{M}{\mathcal{U}_{j_{1}j_{1}j_{2}j_{2}}}}{n}$.
Since $\tilde{c}-c_{nM}\rightarrow 0,$ $\check{c}-c_{nM}\rightarrow 0$ as
$n\rightarrow\infty,$ therefore

$\displaystyle\breve{\varsigma}_{w}^{2}=$
$\displaystyle\sum_{k=1}^{K}\frac{4\phi_{n}^{2}\left(\alpha_{k}\right)\left(\phi_{n}\left(\alpha_{k}\right)-1\right)^{2}}{n}s_{k}^{2}+(\alpha_{x}+1)\left(4c_{nM}^{3}+2c_{nM}^{2}\right)+4\beta_{x}c_{nM}^{3},$

and

$\displaystyle\dfrac{W-(p-M)\breve{\ell}_{W}-\breve{\mu}_{w}}{\breve{\varsigma}_{w}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1),$

then the proof is finished.

###  7.5 Proof of Theorem 4.3

Let $\xi$ be the significance level, and $z_{\xi}$ is the $1-\xi$ quantile of
the standard Gaussian distribution $\Phi$. Since

$\xi=P\left(L>z_{\xi}\varsigma_{l}+p\ell_{l}+\mu_{l}\right),$

for brevity, we denote $L_{0}=p\ell_{l}+\mu_{l}$,
$L_{1}=(p-M)\breve{\ell}_{l}+\breve{\mu}_{l}.$ Therefore, the power to detect
the hypothesis is

$\displaystyle P_{L}=P\left(L>z_{\xi}\varsigma_{l}+L_{0}\right)$
$\displaystyle=P\left(\frac{L-L_{1}}{\breve{\varsigma}_{l}}>\frac{z_{\xi}\varsigma_{l}+L_{0}-L_{1}}{\breve{\varsigma}_{l}}\right)$

Since $\frac{L-L_{1}}{\breve{\varsigma}_{l}}$ is asymptotically normal
distributed, then $P_{L}$ is approxiamte to
$\Phi\left(\frac{L_{1}-L_{0}}{\breve{\varsigma}_{l}}-z_{\xi}\frac{\varsigma_{l}}{\breve{\varsigma}_{l}}\right).$