mathx”17

# Correlation matrix of equi-correlated normal population: fluctuation of the
largest eigenvalue, scaling of the bulk eigenvalues, and stock market

Yohji Akama<EMAIL_ADDRESS>Department of Mathematics, Graduate
School of Science, Tohoku University, Aramaki, Aoba, Sendai, 980-8578, Japan.

((Day Month Year); (Day Month Year))

###### Abstract

Given an $N$-dimensional sample of size $T$ and form a sample correlation
matrix $\mathbf{C}$. Suppose that $N$ and $T$ tend to infinity with $T/N$
converging to a fixed finite constant $Q>0$. If the population is a factor
model, then the eigenvalue distribution of $\mathbf{C}$ almost surely
converges weakly to Marčenko-Pastur distribution such that the index is $Q$
and the scale parameter is the limiting ratio of the specific variance to the
$i$-th variable $(i\to\infty)$. For an $N$-dimensional normal population with
equi-correlation coefficient $\rho$, which is a one-factor model, for the
largest eigenvalue $\lambda$ of $\mathbf{C}$, we prove that $\lambda/N$
converges to the equi-correlation coefficient $\rho$ almost surely. These
results suggest an important role of an equi-correlated normal population and
a factor model in (Laloux et al. Random matrix theory and financial
correlations, _Int. J. Theor. Appl. Finance_ , 2000): the histogram of the
eigenvalue of sample correlation matrix of the returns of stock prices fits
the density of Marčenko-Pastur distribution of index $T/N$ and scale parameter
$1-\lambda/N$. Moreover, we provide the limiting distribution of the largest
eigenvalue of a sample covariance matrix of an equi-correlated normal
population. We discuss the phase transition as to the decay rate of the equi-
correlation coefficient in $N$.

###### keywords:

Portfolio, Marčenko-Pastur distribution, financial correlations, factor model

##  1 Introduction

For risk management and asset allocation, a sample correlation matrix is
essential. One of the pillars of Markowitz portfolio theory (Markowitz, 1959;
Elton _et al._ , 2014) is the study of correlation (or covariance) matrices,
e.g., the optimal weight of each financial asset, given a set of assets with
average returns and risks, so that the overall portfolio offers the best
return for a fixed level of risk or, alternatively, the smallest risk for a
given overall return. For these optimizations, the eigenvalues of
covariance/correlating matrices are important. However, the eigenvalues of a
sample covariance matrix/correlation matrix in finance are dispersed with
noise, so various ways to eliminate noisy eigenvalues from them are studied by
Laloux _et al._ (2000), El Karoui (2008), Ledoit & Wolf (2004, 2012), Ledoit &
Péché (2011), Donoho _et al._ (2018), to cite a few. For this purpose, Laloux
_et al._ (2000) proposed to approximate heuristically the distribution of
noisy eigenvalues of a sample covariance/correlation matrix, with _Marčenko-
Pastur distribution_ (Marčenko & Pastur, 1967) of random matrix theory. Their
idea influenced on the statistical methods on vaccine design (Quadeer _et al._
, 2018). We will discuss the heuristic of Laloux _et al._ (2000)
mathematically, with an _equi-correlated normal population_ , which plays an
important role in various risk models such as _dynamic equicorrelation_ (Engle
& Kelly, 2012) and _constant correlation model_ (Elton _et al._ , 2014).

First of all, we fix the notation. Let
$\bm{X}_{1},\ldots,\bm{X}_{T}\in{{\mathbb{R}}}^{N}$ be a random sample. By the
_data matrix_ , we mean
${\mathbf{X}}=[\bm{X}_{1},\ldots,\bm{X}_{T}]\in{{\mathbb{R}}}^{N\times T}$.
Let ${{1\leq i\leq N}}$ and ${{1\leq t\leq T}}$. The $it$-entry of
${\mathbf{X}}$ is denoted by $x_{it}$, and the $i$-th row by
$\bm{x}_{i}\in{{\mathbb{R}}}^{1\times T}$. For the sample average of
$\bm{x}_{i}$, write $\overline{x}_{i}=T^{-1}{\sum_{t=1}^{T}}x_{it}$. Let
${\overline{\bm{x}}}_{i}:=[\overline{x}_{i},\ldots,\overline{x}_{i}]\in{{\mathbb{R}}}^{1\times
T}$ and
$\overline{\mathbf{X}}:=[\overline{x}_{1},\ldots,\overline{x}_{N}]^{\top}[1,\ldots,1]\in{{\mathbb{R}}}^{N\times
T}$. The _sample correlation matrix_ is, by definition,

$\displaystyle\mathbf{C}$
$\displaystyle={\mathbf{Y}}{\mathbf{Y}}^{\top}\in{{\mathbb{R}}}^{N\times N}\
\text{where}\
{\mathbf{Y}}^{\top}=\left[\frac{(\bm{x}_{1}-{\overline{\bm{x}}}_{1})^{\top}}{\lVert\bm{x}_{1}-{\overline{\bm{x}}}_{1}\rVert},\ldots,\frac{(\bm{x}_{N}-{\overline{\bm{x}}}_{N})^{\top}}{\lVert\bm{x}_{N}-{\overline{\bm{x}}}_{N}\rVert}\right].$

Here, the notation $\lVert.\rVert$ is the Euclidean norm. The _sample
covariance matrix_ is denoted by
$\mathbf{S}=T^{-1}{\mathbf{X}}{\mathbf{X}}^{\top}\in{{\mathbb{R}}}^{N\times
N}$. By a _distribution function_ , we mean a nondecreasing, right-continuous
function $F:{{\mathbb{R}}}\to[0,1]$ such that $\lim_{x\to-\infty}F(x)=0$ and
$\lim_{x\to+\infty}F(x)=1$.

We review the results on eigenvalue distributions from random matrix theory.
The _empirical spectral distribution_ of a symmetric matrix
${\mathbf{P}}\in{{\mathbb{R}}}^{N\times N}$ is, by definition,

$\displaystyle F^{{\mathbf{P}}}(x)=\dfrac{1}{N}\\#\Set{i\,\colon\,{{1\leq
i\leq N}},\ \lambda_{i}({\mathbf{P}})\leq x}\quad(x\in{{\mathbb{R}}})$

where $\lambda_{1}({\mathbf{P}}),\ldots,\lambda_{N}({\mathbf{P}})$ are the
eigenvalues of ${\mathbf{P}}$, indexed in nonincreasing order. According to
(Yao _et al._ , 2015; Bai & Silverstein, 2010, p. 10), _Marčenko-Pastur
distribution with the sample size to dimension ratio index $Q>0$ and scale
parameter $\sigma^{2}\in(0,\,\infty)$_ has a density

$\displaystyle\frac{Q}{2\pi\sigma^{2}x}\sqrt{(\sigma^{2}b_{Q}-x)(x-\sigma^{2}a_{Q})}1_{x\in[\sigma^{2}a_{Q},\,\sigma^{2}b_{Q}]}$
(1)

and has a point mass of value $1-Q$ at the origin if $Q<1$. Here,
$a_{Q}={(1-\sqrt{1/Q})^{2}}$,

$b_{Q}={(1+\sqrt{1/Q})^{2}},$

and $1_{\cdots}$ denotes the indicator function. Marčenko-Pastur distribution
has expectation $\sigma^{2}$. The distribution function of Marčenko-Pastur
distribution of the sample to dimension ratio index $Q$ and scale parameter
$\sigma^{2}$ is denoted by $\mathrm{MP}_{Q,\sigma^{2}}$. We write
$\mathrm{MP}_{Q}$ for $\mathrm{MP}_{Q,1}$. Clearly,
$\mathrm{MP}_{Q,\sigma^{2}}(x)=\mathrm{MP}_{Q}(x/\sigma^{2})$ for all
$x\in{{\mathbb{R}}}$. We say a sequence $(F_{i})_{i}$ of distribution
functions _converges weakly_ to a function $F$, if $F_{i}(x)\to F(x)$ for
every $x\in{{\mathbb{R}}}$ where $F$ is continuous.

######  Proposition 1.1.

Let $\set{x_{it}\colon\,i,t\geq 1}$ be a double array of _i.i.d._ random
variables with $\operatorname{E}|x_{11}|^{2}=\sigma^{2}\in(0,\,\infty)$.
Suppose ${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$. Then, almost surely,

  1. 1.

$F^{\mathbf{S}}$ converges weakly to $\mathrm{MP}_{Q,\sigma^{2}}$. (Yao _et
al._ , 2015, p. 12)

  2. 2.

$F^{\mathbf{C}}$ converges weakly to $\mathrm{MP}_{Q}$. (Jiang, 2004, Theorem
1.2)

However, the eigenvalues of financial correlation matrices distribute
similarly as the random matrix theory suggests, but somehow differently.
Laloux _et al._ (2000) considered a sample correlation matrix $\mathbf{C}$
“associated to the time series of the different stocks of the S&P500 (or other
major markets)” where $N$ is the number of assets and $T$ is the number of
observations. Laloux _et al._ (2000) found that the histogram of the
eigenvalues of the financial, sample correlation matrix $\mathbf{C}$ fits not
the density of $\mathrm{MP}_{T/N}$, but a _scaled_ density, specifically, the
density of

$\displaystyle\mathrm{MP}_{\frac{T}{N},\,1-\frac{\lambda_{1}(\mathbf{C})}{N}}.$
(2)

In addition to the fitting result, for the first eigenvalue
$\lambda_{1}(\mathbf{C})$ of $\mathbf{C}$, Laloux et al. remarked the
following:

> The corresponding eigenvector is, as expected, the “market” itself, i.e., it
> has roughly equal components on all the $N$ stocks.

Laloux _et al._ (2000) interpreted the support of the density function of the
fitting Marčenko-Pastur distribution, as the interval of noisy eigenvalues of
the sample correlation matrix $\mathbf{C}$. Based on the findings of Laloux
_et al._ (2000), in order to improve risk management, Bun _et al._ (2017)
proposed “ _eigenvalue clipping_ : all eigenvalues in bulk of the fitting
Marčenko-Pastur spectrum are deemed as noise and thus replaced by a constant
value whereas the principal components outside of the bulk (the spikes) are
left unaltered.” As portfolios of hundreds or thousands of assets becomes the
target of research, many approaches have been proposed to deal with the
problem of dimensionality. By the limiting regime ${{N,T\to\infty,\ {T/N\to
Q}}}\in(0,\infty)$ of Proposition 1.1, we would like to understand
mathematically why the heuristic of Laloux _et al._ (2000) is meaningful.

Let us hypothesize about the population of the stock prices Laloux et al.
considered, from their remark on the eigenvector that “has roughly equal
components on all the $N$ stocks.” It is comparable to an eigenvector
$[1,\ldots,1]^{\top}$ corresponding to the largest eigenvalue
$\lambda_{1}({\mathbf{\Sigma}}_{\rho})$ of an _equi-correlation matrix_

$\displaystyle{\mathbf{\Sigma}}_{\rho}=\begin{bmatrix}1&\rho&\cdots&\rho\\\
\rho&1&\cdots&\rho\\\ \vdots&\vdots&&\vdots\\\
\rho&\rho&\cdots&1\end{bmatrix}\in{{\mathbb{R}}}^{N\times
N}\qquad(0\leq\rho<1),$

where

$\displaystyle\lambda_{1}({\mathbf{\Sigma}}_{\rho})=(N-1)\rho+1\geq\lambda_{2}({\mathbf{\Sigma}}_{\rho})=\cdots=\lambda_{N}({\mathbf{\Sigma}}_{\rho})=1-\rho.$
(3)

SenGupta (1988) studied hypothesis tests on $\rho=0$ vs. $\rho>0$. Ishii _et
al._ (2021) developed nonparametric tests of the high-dimensional covariance
structures such as a scaled identity matrix, a diagonal matrix and
${\mathbf{\Sigma}}_{\rho}$ for $N>T$ cases.

In particular, when $\rho>0$, $\lambda_{1}({\mathbf{\Sigma}}_{\rho})$ is
_unbounded_ with respect to $N$, and thus there is a ‘spectral gap’ between
$\lambda_{1}({\mathbf{\Sigma}}_{\rho})=(N-1)\rho+1$ and
$\lambda_{2}({\mathbf{\Sigma}}_{\rho})=1-\rho$. We presume that the population
is a normal population with the population correlation matrix
${\mathbf{\Sigma}}_{\rho}$.

Then, for an equi-correlated normal population, we established the following
scaling theorem for the limiting distribution of the _bulk eigenvalues_ of the
sample correlation matrix $\mathbf{C}$:

######  Proposition 1.2 (Akama & Husnaqilati (2022)).

Let $\mathbf{C}$ be a sample correlation matrix formed from an $N$-dimensional
normal population such that the population correlation matrix is
${\mathbf{\Sigma}}_{\rho}$ with deterministic $\rho\in[0,\,1)$. Suppose
${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$. Then, almost surely, the
empirical spectral distribution $F^{\mathbf{C}}$ converges weakly to

$\mathrm{MP}_{Q,\,1-\rho}.$

It is worth noting that the limiting spectral distribution (LSD)
$\mathrm{MP}_{Q,\,1-\rho}$ is similar to Marčenko-Pastur distribution (2)
$\mathrm{MP}_{T/N,\,1-\lambda_{1}(\mathbf{C})/N}$ where the histogram of the
eigenvalues of a financial correlation matrix fits to the density of (2). The
index $Q$ and the scale parameter $1-\rho$ correspond to $T/N$ and
$1-\lambda_{1}(\mathbf{C})/N$, respectively.

By comparing (2) to Proposition 1.2, we will show that the largest eigenvalue
$\lambda_{1}(\mathbf{C})$ of the sample correlation matrix $\mathbf{C}$
divided by the order $N$ of $\mathbf{C}$ converges almost surely to the equi-
correlation coefficient $\rho$, as follows. Hereafter,
${\stackrel{{\scriptstyle a.s.}}{{\to}}}$ and ${\stackrel{{\scriptstyle
D}}{{\to}}}$ mean the almost sure convergence and the convergence in
distribution respectively. Unless otherwise stated, the limiting regime is
${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$.

######  Theorem 1.3.

Suppose the assumptions of Proposition 1.2. Then,

  1. 1.

$\lambda_{1}(\mathbf{C})\big{/}{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$.

  2. 2.

If, moreover, the population is $\mathrm{N}_{N}(\bm{0},\
{\mathbf{\Sigma}}_{\rho})$ and $\rho>0$, then

$\displaystyle\frac{\lambda_{1}(\mathbf{S})-\tau}{\varsigma}\
{\stackrel{{\scriptstyle D}}{{\to}}}\ \mathrm{N}(0,1)$

where

$\displaystyle\tau=\frac{\left((N-1)\rho+1\right)\left(\left(1+(T-1)N\right)\rho+N-1\right)}{NT\rho},\quad\varsigma=\frac{(N-1)\rho+1}{\sqrt{2T}}.$

From the first assertion, i.e., the consistency of
$\lambda_{1}(\mathbf{C})/N$, of Theorem 1.3, Proposition 1.2 suggests that an
equi-correlated normal population plays a role in Laloux et al.’s fitting
result to the density of (2):

######  Corollary 1.4.

Assume the assumptions of Proposition 1.2. Then,

$\displaystyle
F^{\mathbf{C}}(x)-\mathrm{MP}_{\frac{T}{N},\,1-\frac{\lambda_{1}(\mathbf{C})}{N}}(x)\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0\quad(x\neq 0).$

Once the largest eigenvalue $\lambda_{1}(\mathbf{C})$ of $\mathbf{C}$ is
found, Corollary 1.4 can suggest the support of the density of the fitting
Marčenko-Pastur distribution to clean up (“eigenvalue clipping”), and then we
can eliminate the eigenvalues in the support to let a small, but important
eigenvalue (e.g., Markowitz solution) show up. Moreover,
$\lambda_{1}(\mathbf{C})$ has been concerned by Friedman & Weisberg (1981).

The rest of the paper is organized as follows. In the next section, to
generalize Proposition 1.2, we consider suitable factor models. Then, we
establish the limiting spectral distribution of the sample correlation matrix
$\mathbf{C}$ generated from such a factor model. In Section 3, for returns of
S&P stock prices (2012-1-4/2021-12-31), we compute the largest eigenvalues of
financial correlation matrices $\mathbf{C}$ divided by the order, and estimate
the time average $\overline{\rho}$ of equi-correlation coefficient by GJR
GARCH (Glosten _et al._ , 1993) with the correlation structure being
‘equicorrelation’ (Engle & Kelly, 2012). We show that
$\lambda_{1}(\mathbf{C})/N$ predicts very well $\overline{\rho}$. In Section
4, for a nonnormal population with equicorrelation coefficient $\rho=\rho_{N}$
decaying in $N$, we review the consistency and the asymptotic normality of
$\lambda_{1}(\mathbf{S})$, by citing Yata & Aoshima (2009). Then, for the
limiting distribution of an appropriate scaling of a translation of
$\lambda_{1}(\mathbf{S})$ of equi-correlated normal population, we discuss the
phase transition from the normal population to _Tracy-Widom distribution_
(Baik _et al._ , 2005), as to the decay rate of $\rho_{N}$. The proofs of
Theorem 1.3, Corollary 1.4, and Theorem 2.3 are postponed in Section 5,
Section 6, and Section 7, respectively, for the sake of the presentation.

##  2 From equi-correlated normal population to factor models in limit

Proposition 1.2 was proved by the following well-known fact (Fan & Jiang,
2019, (3.21)): Assume that
$\bm{X}_{1},\ldots,\bm{X}_{T}\stackrel{{\scriptstyle\mbox{{i.\@i.\@d.\@}}}}{{\sim}}\mathrm{N}_{N}(\bm{0},\,{\mathbf{\Sigma}}_{\rho})$,
$0\leq\rho<1$, and
$[\bm{X}_{1},\ldots,\bm{X}_{T}]=[x_{it}]_{it}\in{{\mathbb{R}}}^{N\times T}$.
Then, there are independent standard normal random variables ${{f}}_{t}$,
$e_{it}$ $({{1\leq i\leq N}},\ {{1\leq t\leq T}})$ such that

$\displaystyle x_{it}=\sqrt{\rho}{{f}}_{t}+\sqrt{1-\rho}e_{it}.$ (4)

Actually, an equi-correlated normal population is a _factor model_ (Bai, 2003;
Mulaik, 2009) consisting of 1 factor. See Example 2.2. Beside this 1-factor
model, a 3-factor model and a 5-factor model are employed by Fama & French
(1993, 2015) to describe stock returns in asset pricing and portfolio
management.

Bai (2003) studied an inferential theory for a general factor model of large
dimensions to which classical factor analysis does not apply, by using
limiting regime $N,T\to\infty,\sqrt{N}/T\to 0$. He derived the limiting
distributions of the estimated factors and estimated factor loadings, the
asymptotic normality of estimated common components, and their convergence
rates. The convergence rate of the estimated factors and factor loadings can
be faster than that of the estimated common components. These achievements are
obtained under liberal conditions that allow for _correlations_ and
heteroskedasticities. Stronger results are obtained when the idiosyncratic
errors are serially uncorrelated and homoskedastic. A necessary and sufficient
condition for consistency of the estimated factors is derived for large $N$
but fixed $T$.

We will study a factor model in the limiting regime $N,T\to\infty,\ T/N\to Q$.
First, we recall a factor model (Bai, 2003; Mulaik, 2009):

######  Definition 2.1.

For a nonnegative integer $K$, by a _$K$ -factor model_, we mean a data matrix

$\left[x_{it}\right]_{it}\in{{\mathbb{R}}}^{N\times T}$

such that

$\displaystyle x_{it}=\mu_{i}+\bm{\ell}_{i}{{\bm{f}}}_{t}+e_{it}\quad(1\leq
i\leq N,\ 1\leq t\leq T),$ $\displaystyle\mbox{$\mu_{i}$ is a deterministic
constant,}\qquad\qquad\qquad\qquad\qquad\
\qquad\qquad\qquad\qquad(\mbox{\emph{mean}})$
$\displaystyle\mbox{${\bm{\ell}}_{i}=[\ell_{i1},\ldots,\ell_{iK}]$ is a
$K$-dimensional row vector,}\qquad\qquad\qquad(\mbox{\emph{factor loadings}})$
$\displaystyle\mbox{${{\bm{f}}}_{t}=[f_{t1},\ldots,f_{tk}]^{\top}$ is a
$K$-dimensional column vector,}\qquad\quad(\mbox{\emph{common factors}})$
$\displaystyle\mbox{$f_{tk}$ $(1\leq t\leq T,\ 1\leq k\leq K)$ are centered,
i.i.d. with $\operatorname{Var}f_{t1}=1$},$ $\displaystyle\mbox{$e_{it}$
$(1\leq t\leq T)$ are centered, i.i.d., for each $i$ $(1\leq i\leq N)$},$
$\displaystyle\mbox{$f_{tk}$ and $e_{it}$ $(1\leq i\leq p,\ 1\leq t\leq T,\
1\leq k\leq K)$ are independent},\ \mbox{and}$
$\displaystyle\psi_{i}:=\operatorname{Var}e_{it}<\infty\quad\quad\quad\quad\quad\quad\quad\quad\quad\qquad\qquad\qquad\qquad\qquad(\mbox{\emph{specific
variance}}).$

$e_{it}$’s are called _idiosyncratic components_. $K$ is the number of
factors.

######  Example 2.2.

The decomposition (4) is a factor model such that $K=1$, $\mu_{i}:=0$,
$\ell_{i1}=\sqrt{\rho}$, $\psi_{i}:=1-\rho$, and both of $f_{t1}$ and $e_{it}$
are independent and normal.

$e_{it}$ $(i,t\geq 1)$ are _i.i.d._ , and ${\bm{\ell}}_{i}\to\bm{\ell}$ for
some $K$-dimensional row vector $\bm{\ell}$. In this case, define

$\displaystyle\sigma_{\infty}^{2}:=\lim_{i\to\infty}\operatorname{Var}x_{it}=\lVert\bm{\ell}\rVert^{2}+\psi_{1}<\infty,\qquad\rho:=\frac{\lVert\bm{\ell}\rVert^{2}}{\lVert\bm{\ell}\rVert^{2}+\psi_{1}}.$

Then, $1-\rho$ is the portion of the specific variance to the limiting total
variance $\sigma_{\infty}^{2}$ of a variable $x_{it}$ $(i\to\infty)$.

The following generalizes Proposition 1.2. The LSD of a sample correlation
matrix $\mathbf{C}$ generated from a factor model is as follows:

######  Theorem 2.3.

Given a factor model. Suppose ${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$
and Assumption 2. Then, almost surely, $F^{\mathbf{C}}$ converges weakly to
$\mathrm{MP}_{Q,\,1-\rho}$.

By Example 2.2, this generalizes Proposition 1.2. Theorem 2.3 is proved in
Section 7.

##  3 The largest eigenvalue of financial correlation matrix

For a dataset of returns of $N$ S&P500 stocks (or other major markets) for $T$
trading days, Laloux _et al._ (2000) fitted the histogram of the eigenvalues
of the correlation matrix $\mathbf{C}$, to the density function of a scaled
Marčenko-Pastur distribution.

Naturally, the return of a stock is more correlated with the return of a stock
of the same industry classification sector, than with the return of a stock of
a different industry classification sector. We consider the 11 _global
industry classification standard_ (GICS) sectors of S&P500 stocks. For each
GICS, for the dataset of the stock returns for the period
2012-01-04/2021-12-31, the time series of the equi-correlation coefficient is
computed by GJR GARCH (Glosten _et al._ , 1993) with the correlation structure
being _dynamic equicorrelation_ (Engle & Kelly, 2012) (We abbreviate this
model to ‘GJR GARCH-DECO’). The program is due to Candila (2021).

However, the computation time of the time series of the equi-correlation
coefficient by GJR GARCH-DECO is large, although we can quickly compute the
estimator $\lambda_{1}(\mathbf{C})/N$ of the equi-correlation coefficient by a
power method. For each dataset, we compare the time series to
$\lambda_{1}(\mathbf{C})/N$ of the correlation matrix $\mathbf{C}$.

Figure 1: Time series of $\rho$ and $\lambda_{1}(\mathbf{C})/N$ (red line) of
the stock returns of the 12 datasets of Table 1.

Figure 1 is the time series of equi-correlations for the 12 datasets. All the
time series rise around March 2020, by the world-wide sudden crash after
growing uncertainty due to the COVID-19 (Coronavirus disease 2019) pandemic.
The time series of the equi-correlation coefficient of Energy sector kept
high, even after the time series of the equi-correlation of the other 10
sectors of GICS as well as the total decrease. According to Figure 1, for each
dataset, the time series is mostly smaller than $\lambda_{1}(\mathbf{C})/N$.
Let us consider the time average $\overline{\rho}$ of equi-correlation
coefficients estimated by GJR GARCH-DECO. Table 1 is the list of GICS, $N/T$,
$\lambda_{1}(\mathbf{C})/N$, $N$, and $\overline{\rho}$, for the period
2012-01-04/2021-12-31 ($T=2516$=trading days $-1$). The rows of Table 1 are
ordered in the increasing order of $\lambda_{1}(\mathbf{C})/N$.

No | GICS | $N/T$ | $\lambda_{1}(\mathbf{C})/N$ | $N$ | $\overline{\rho}$
---|---|---|---|---|---
1 | Communication Services | .0076 | .3571 | 19 | .2764
2 | Consumer Discretionary | .0207 | .3843 | 52 | .2967
3 | Health Care | .0187 | .3948 | 47 | .334
4 | Consumer Staples | .0091 | .4302 | 23 | .3606
5 | Information Technology | .0246 | .4648 | 62 | .3894
6 | Industrials | .0258 | .4985 | 65 | .4241
7 | Materials | .0095 | .499 | 24 | .4104
8 | Real estate | .0119 | .5819 | 30 | .5016
9 | Financials | .025 | .6086 | 63 | .5159
10 | Energy | .0064 | .6872 | 16 | .5954
11 | Utilities | .0111 | .6897 | 28 | .621
12 | The total | .1705 | .3819 | 429 | .2939
Table 1: The stock return datasets of 11 GICS’s and the total
(2012-01-04/2021-12-31), the largest eigenvalue of the correlation matrices
and the time averages of equicorrelation coefficient. $N/T$,
$\lambda_{1}(\mathbf{C})/N$ and $\overline{\rho}$ are rounded.

The last row of Table 1 is for the union dataset of the 11 GICS datasets. The
union dataset is more ‘heterogeneous’ than the 11 GICS datasets. It has
$\lambda_{1}(\mathbf{C})/N$ and $\overline{\rho}$ smaller than the
corresponding $\lambda_{1}(\mathbf{C})/N$ and $\overline{\rho}$ for most of
GICS. It means that correlations coefficients between different GICS’s are
very small. Figure 2 are the heat maps with hierarchical clustering algorithms
for the correlation matrices $\mathbf{C}$ of the 12 datasets. According to the
last heat map of Figure 2, the correlation matrix of the returns of S&P500
stocks has diagonal block structures as well as an almost dominant background.

(a) Communication Services

(b) Consumer Discretionary

(c) Health Care

(d) Consumer Staples

(e) Information Technology

(f) Industrials

(g) Materials

(h) Real Estate

(i) Financials

(j) Energy

(k) Utilities

(l) The total

Figure 2: The correlation matrices of the stock returns of the 12 datasets of
Table 1 and the total.

Although Figure 2 shows the various structures of $\mathbf{C}$ for the 12
datasets, $\lambda_{1}(\mathbf{C})/N$ predicts well the time average
$\overline{\rho}$ of the time series of equi-correlation coefficients computed
by GJR GARCH-DECO with much time. A regression analysis (Figure 3) for Table 1
shows:

$\displaystyle\overline{\rho}\sim
0.98488\frac{\lambda_{1}(\mathbf{C})}{N}-0.07190\quad(\mbox{adjusted $R^{2}$
being 0.9916}).$

Figure 3: The regression analysis of $\overline{\rho}$ by
$\lambda_{1}(\mathbf{C})/N$.

##  4 Phase transition of limiting distribution of largest eigenvalue

In our setting, all the correlation coefficients among $N$ variables being the
positive constant $\rho$. It is, however, somehow unnatural in the limit
$N\to\infty.$ We assume that

$N>T$

and the equi-correlation coefficient $\rho$ converges to 0 from above as
$N\to\infty$.

For this setting with $N>T$, the consistency
$\lambda_{1}(\mathbf{S})/N\to\rho$ of the largest eigenvalue
$\lambda_{1}(\mathbf{S})$ and the asymptotic normality of
$\lambda_{1}(\mathbf{S})$ for a nonnormal population follows from Yata &
Aoshima (2009). Moreover, the limiting distribution of
$\lambda_{1}(\mathbf{S})$ for an equi-correlated normal population seems to
have a phase transition depending on the equi-correlation coefficient $\rho$,
as the limiting distribution of the largest eigenvalue of $\mathbf{S}$ formed
by a spiked population model does so with respect to the largest eigenvalue of
the population covariance matrix (Baik _et al._ , 2005).

###  4.1 Consistency and asymptotic normality of $\lambda_{1}(\mathbf{S})$ of
nonnormal population

In (Yata & Aoshima, 2009), the consistency and the asymptotic normality of
$\lambda_{1}(\mathbf{S})$ are proved without assuming the normality of the
population.

######  Proposition 4.1.

Let a data matrix ${\mathbf{X}}\in{{\mathbb{R}}}^{N\times T}$ be
$[\bm{X}_{1},\ldots,\bm{X}_{T}]$ where $\bm{X}_{t}$ $(1\leq t\leq T)$ are
centered, i.i.d. and $N>T$. Suppose that the population covariance matrix
${\mathbf{\Sigma}}$ is
$\mathbf{H}\operatorname{diag}(\lambda_{1},\ldots,\lambda_{N})\mathbf{H}^{\top}$
for some orthogonal matrix $\mathbf{H}$,

$\displaystyle\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\geq 0,$
$\displaystyle\lambda_{i}=c_{i}N^{\delta_{i}}\ (i=1,\ldots,m),\ \mbox{and}$
(5) $\displaystyle\mbox{$\lambda_{j}$ are constants $(j=m+1,\ldots,N)$}.$

Here $m$, $c_{i}(>0)$ and $\delta_{i}$
$(\delta_{1}\geq\cdots\geq\delta_{m}>0)$ are constants.

Let
$\mathbf{Z}:=[\bm{Z}_{1},\ldots,\bm{Z}_{N}]:=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{N})^{-1/2}\mathbf{H}^{\top}{\mathbf{X}}$.
Assume that $\bm{Z}_{1},\ldots,\bm{Z}_{N}$ are independent, the fourth moments
of the entries of $\mathbf{Z}$ are uniformly bounded, and no $\bm{Z}_{i}$ is
$\bm{0}$. Moreover suppose

  1. (i)

$N,T\to\infty$ for $i$ such that $\delta_{i}>1$; and

  2. (ii)

$N,T\to\infty$ and $N^{2-2\delta_{i}}/T\to 0$ for $i$ such that
$\delta_{i}\in(0,\,1]$.

Then, we have

  1. 1.

$\lambda_{1}(\mathbf{S})/\lambda_{1}=1+o_{p}(1)$. (Yata & Aoshima, 2009,
Theorem 1)

  2. 2.

Assume also that the largest $m$ eigenvalues of ${\mathbf{\Sigma}}$ are
distinct and $\operatorname{Var}(z_{ij}^{2})=M_{i}\ (<\infty)$. Then,
$\sqrt{T/M_{i}}(\lambda_{1}(\mathbf{S})/\lambda_{1}-1)\mathrel{\stackrel{{\scriptstyle
D}}{{\to}}}\mathrm{N}(0,\,1)$. (Yata & Aoshima, 2009, Theorem 2)

The condition (ii) is relieved for (Yata & Aoshima, 2009, Theorem 2) if an
independence condition is further assumed.

######  Proposition 4.2 ((Yata & Aoshima, 2009, Corollary 1)).

In the assertion (2) of Proposition 4.1, assume further that $z_{ij}$, $1\leq
i\leq N(1\leq t\leq T)$ are independent. Then, it holds that
$\lambda_{1}(\mathbf{S})/\lambda_{1}=1+o_{p}(1)$ if the condition (i) and

> (iii) $N,T\to\infty$ and $N^{1-2\delta_{i}}/T\to 0$ for $i$ such that
> $\delta_{i}\in(0,\,1]$.

By (3), ${\mathbf{\Sigma}}_{\rho}$ does not meet the condition (5). According
to Kazuyoshi Yata, the conclusions of (Yata & Aoshima, 2009, Theorem 1) and
(Yata & Aoshima, 2009, Theorem 2) indeed hold for
${\mathbf{\Sigma}}={\mathbf{\Sigma}}_{\rho}$. Therefore, we get the following:

######  Theorem 4.3.

Suppose that a population is possibly nonnormal, the population covariance
matrix is ${\mathbf{\Sigma}}_{\rho}$ $(\rho>0)$, and $N,T\to\infty,{T/N\to
Q}\in(0,\,\infty)$. Assume

  1. 1.

the setting of (Yata & Aoshima, 2009, Theorem 1) and
$\lim_{N\to\infty}N^{1/2}\rho=\infty$; or

  2. 2.

the setting of (Yata & Aoshima, 2009, Corollary 1) and
$\lim_{N\to\infty}N\rho=\infty$.

Then, we have ${\lambda_{1}(\mathbf{S})}/{N}\mathrel{\stackrel{{\scriptstyle
P}}{{\to}}}\rho$ and the asymptotic normality of $\lambda_{1}(\mathbf{S})$.

###  4.2 Phase transition of limiting distribution of
$\lambda_{1}(\mathbf{S})$ for normal population as to decay rate of equi-
correlation coefficient

The limiting distribution of an appropriate scaling of a translation of the
largest eigenvalue $\lambda_{1}(\mathbf{S})$ of a sample covariance matrix
$\mathbf{S}$ is Tracy-Widom distribution if the equi-correlation coefficient
$\rho$ is 0, by (Johnstone, 2001; Soshnikov, 2002), and a normal distribution
if $\rho$ is a positive constant by Theorem 5.2. In view of Theorem 4.3 (2),
we turn our attention to the case $\lim_{N\to\infty}N\rho<\infty$.

First, we recall (Baik _et al._ , 2005, Theorem 1.1 and Corollary 1.1).

######  Proposition 4.4.

Assume $Q>1$ and the population is
$\mathrm{N}_{N}(\bm{\mu},\,{\mathbf{\Sigma}})$, and
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{r}\geq\lambda_{r+1}=\cdots=\lambda_{N}=1$
be the eigenvalues of ${\mathbf{\Sigma}}$.

  * (a)

Suppose $0\leq k\leq r$, $\lambda_{1}=\cdots=\lambda_{k}=1+1/\sqrt{Q}$, and
$\set{\lambda_{k+1},\ldots,\lambda_{r}}\subset(0,\,1+1/\sqrt{Q})$. Then,

$\displaystyle\lambda_{1}(\mathbf{S})\mathrel{\stackrel{{\scriptstyle
P}}{{\to}}}{(1+\sqrt{1/Q})^{2}},\ \mbox{and}$
$\displaystyle\operatorname{P}\left(\left(\lambda_{1}(\mathbf{S})-{(1+\sqrt{1/Q})^{2}}\right)\cdot\frac{\sqrt{Q}}{(1+\sqrt{Q})^{4/3}}T^{2/3}\leq
x\right)\to F_{\mathrm{TW},k}(x),$ (6)

where $F_{\mathrm{TW},k}(x)$ is the distribution function of a generalization
of the Tracy-Widom distribution (Baik _et al._ , 2005, Definition 1.1).

  * (b)

Suppose $1\leq k\leq r$, $\lambda_{1}=\cdots=\lambda_{k}>1+1/\sqrt{Q}$, and
$\set{\lambda_{k+1},\ldots,\lambda_{r}}\subset(0,\lambda_{1})$. Then,

$\displaystyle\lambda_{1}(\mathbf{S})\stackrel{{\scriptstyle
P}}{{\to}}\lambda_{1}+\frac{Q^{-1}\lambda_{1}}{\lambda_{1}-1},\ \mbox{and}$
$\displaystyle\operatorname{P}\left(\left(\lambda_{1}(\mathbf{S})-\left(\lambda_{1}+\frac{Q^{-1}\lambda_{1}}{\lambda_{1}-1}\right)\right)\cdot\frac{\sqrt{T}}{\sqrt{\lambda_{1}^{2}-Q^{-1}\lambda_{1}^{2}/(\lambda_{1}-1)^{2}}}\leq
x\right)\to F_{\mathrm{G},k}(x),$ (7)

where $F_{\mathrm{G},k}(x)$ is the distribution function of a generalization
of the Gaussian distribution (Baik _et al._ , 2005, Definition 1.2).

This proposition suggests the following phase transition: For a normal
population, if $\lim_{N\to\infty}N\rho>1/\sqrt{Q}$, then the largest
eigenvalue $\lambda_{1}(\mathbf{S})$ satisfies the asymptotic normality in a
sense of (6), and if $\lim_{N\to\infty}N\rho<1/\sqrt{Q}$, then an appropriate
scaling of a translation of $\lambda_{1}(\mathbf{S})$ converges to the Tracy-
Widom distribution $G_{1}$ in a sense of (7). This holds for
$\lim_{N\to\infty}N\rho=\infty$ by Theorem 4.3.

The detail of this phase transition will be studied elsewhere.

##  5 Proof of Theorem 1.3

We also consider centered (noncentered, resp.) variant of the sample
covariance matrix $\mathbf{S}\in{{\mathbb{R}}}^{N\times N}$ (the sample
correlation matrix $\mathbf{C}\in{{\mathbb{R}}}^{N\times N}$, resp.):

######  Definition 5.1.

Let ${\mathbf{\tilde{Y}}}$ be
$\operatorname{diag}\left(\lVert\bm{x}_{1}\rVert^{-1},\ldots,\lVert\bm{x}_{N}\rVert^{-1}\right){\mathbf{X}}\in{{\mathbb{R}}}^{N\times
T}$. Define the following three matrices:

$\displaystyle{\mathbf{\tilde{C}}}\ \ $
$\displaystyle:={\mathbf{\tilde{Y}}}{\mathbf{\tilde{Y}}}^{\top}\in{{\mathbb{R}}}^{N\times
N}$ $\displaystyle\mbox{(\emph{noncentered sample correlation matrix})},$
$\displaystyle\mathbf{E}\ \ $
$\displaystyle:=T^{-1/2}({\mathbf{X}}-\overline{\mathbf{X}})\in{{\mathbb{R}}}^{N\times
T},$ $\displaystyle\mathbf{S}^{\circ}\ $
$\displaystyle:=\mathbf{E}\mathbf{E}^{\top}\in{{\mathbb{R}}}^{N\times N}$
$\displaystyle\mbox{(\emph{centered sample covariance matrix})}.$

To prove Theorem 1.3 (1), we will prove:

######  Theorem 5.2.

Assume that
$\bm{X}_{1},\ldots,\bm{X}_{T}\mathrel{\stackrel{{\scriptstyle\mbox{\emph{i.\@i.\@d.\@}}}}{{\sim}}}\mathrm{N}_{N}(\bm{\mu},\,\bm{\Delta}{\mathbf{\Sigma}}_{\rho}\bm{\Delta})$,
$\bm{\mu}\in{{\mathbb{R}}}^{N}$ is deterministic, $\bm{\Delta}$ is
deterministic, nonsingular diagonal, and $\rho\in[0,\,1)$ is deterministic.
Suppose $({\mathbf{P}}=\mathbf{S}$, $\bm{\mu}=\bm{0}$,
$\bm{\Delta}=\mathbf{I})$, $({\mathbf{P}}=\mathbf{S}^{\circ}$,
$\bm{\Delta}=\mathbf{I})$, $({\mathbf{P}}={\mathbf{\tilde{C}}}$,
$\bm{\mu}=\bm{0})$, or ${\mathbf{P}}=\mathbf{C}$. Then,
${\lambda_{1}({\mathbf{P}})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$.

######  Proof 5.3.

First consider the case where ${\mathbf{P}}=\mathbf{S}$, $\bm{\mu}=\bm{0}$,
and $\bm{\Delta}=\mathbf{I}$. Suppose $\rho=0$. Then, all the entries of
${\mathbf{X}}$ are standard normal, so they are independent and
$\operatorname{E}|x_{11}|^{4}<\infty$. Thus,

$\displaystyle\lim_{N,T\to\infty,{T/N\to
Q}}\lambda_{1}(\mathbf{S})=b_{Q}\quad(a.s.)$

by (Bai & Yin, 1993, Theorem 2). Hence, Theorem 5.2 (1) holds for this case
when $\rho=0$.

Then, we prove $\lambda_{1}(\mathbf{S})/N\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ for $\rho>0$. For this, we recall (Merlevède _et al._ ,
2019, Proposition 2.1), which is due to (Ledoit & Wolf, 2018, Proposition 7.3)
and (Cai _et al._ , 2017, Theorem 2.1) according to Merlevède _et al._ (2019).
Following (Van der Vaart, 2000, p. 8), we say a sequence $(F_{N})_{N}$ of
distribution functions is _tight_ on ${{\mathbb{R}}}^{+}$, if for every
$\epsilon\in(0,\,1]$, there is a constant $M\geq 0$ such that

$\sup_{N}F_{N}(M)>1-\epsilon.$

######  Proposition 5.4 (Merlevède _et al._ (2019, Proposition 2.1)).

Assume

  1. 1.

$\mathbf{S}=T^{-1}{{\mathbf{\Sigma}}}^{1/2}{\mathbf{G}}{\mathbf{G}}^{\top}{{\mathbf{\Sigma}}}^{1/2}\in{{\mathbb{R}}}^{N\times
N}$, the entries of ${\mathbf{G}}\in{{\mathbb{R}}}^{N\times T}$ are
independent, standard normal, and
${{\mathbf{\Sigma}}}\in{{\mathbb{R}}}^{N\times N}$ is positive definite,
symmetric, and deterministic.

  2. 2.

A sequence $\left(F^{{{\mathbf{\Sigma}}}}\right)_{N}$ of empirical spectral
distributions is tight on ${{\mathbb{R}}}^{+}$, and
$\lim_{N\to\infty}\lambda_{1}({{\mathbf{\Sigma}}})=\infty$.

  3. 3.

${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\infty)$.

Then,
${\lambda_{1}(\mathbf{S})}/{\lambda_{1}({{\mathbf{\Sigma}}})}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}1.$

The sequence $\left(F^{{\mathbf{\Sigma}}_{\rho}}\right)_{N}$ of the empirical
spectral distribution function of
${\mathbf{\Sigma}}_{\rho}\in{{\mathbb{R}}}^{N\times N}$ is a tight sequence on
${{\mathbb{R}}}^{+}$. To see it, consider $\epsilon\in(0,\,1]$ and
$M=(\epsilon^{-1}-1)\rho+1>0$. For every integer $N>M$,
$\lambda_{1}({\mathbf{\Sigma}}_{\rho})>M$, so
$F^{{\mathbf{\Sigma}}_{\rho}}(M)=1$. Hence,
$\sup_{N}F^{{\mathbf{\Sigma}}_{\rho}}(M)\geq 1-\epsilon$. Therefore, we have
the tightness of the sequence.

Then, by Proposition 5.4 with
${{\mathbf{\Sigma}}}:={\mathbf{\Sigma}}_{\rho}\in{{\mathbb{R}}}^{N\times N}$,
${\lambda_{1}(\mathbf{S})}\big{/}\left((N-1)\rho+1\right)={\lambda_{1}(\mathbf{S})}\big{/}{\lambda_{1}({\mathbf{\Sigma}}_{\rho})}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}1.$ Thus, we have established
$\lambda_{1}(\mathbf{S})/N\mathrel{\stackrel{{\scriptstyle a.s.}}{{\to}}}\rho$
for every $\rho\geq 0$.

Now, we consider the case where ${\mathbf{P}}=\mathbf{S}^{\circ}$ and
$\bm{\Delta}=\mathbf{I}$. We will derive
${\lambda_{1}(\mathbf{S}^{\circ})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ for every $\rho\geq 0$, by using the following:

######  Proposition 5.5 (Bai & Yin (1993, Lemma 2)).

Let $\set{y_{kj}\colon\,k,j\geq 1}$ be a double array of _i.i.d._ random
variables and let $\alpha>{1}/{2}$, $\beta\geq 0$ and $M>0$ be constants.
Then,

$\displaystyle\max_{j\leq
MT^{\beta}}\left|\sum_{k=1}^{T}\frac{y_{kj}-m}{T^{\alpha}}\right|\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0\quad(T\to\infty)$ $\displaystyle\iff$
$\displaystyle\operatorname{E}|y_{11}|^{\frac{1+\beta}{\alpha}}<\infty\;\&\;m=\begin{cases}\displaystyle\operatorname{E}y_{11},&(\alpha\leq
1),\\\ \displaystyle\text{\rm any},&(\alpha>1).\end{cases}$

Proposition 5.5 is also used to prove Theorem 2.3 in Section 7.

The centered sample covariance matrix $\mathbf{S}^{\circ}$ is invariant with
respect to the shifting of variables. By this invariance, we can assume
$\bm{\mu}=\bm{0}.$ It suffices to assure

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\sqrt{\frac{\lambda_{1}(\mathbf{S}^{\circ})}{N}}-\sqrt{\frac{\lambda_{1}(\mathbf{S})}{N}}\right|=0\quad(a.s.).$

By the triangle inequality for the spectral norm,

$\displaystyle\left|\sqrt{\lambda_{1}(\mathbf{S}^{\circ})}-\sqrt{\lambda_{1}(\mathbf{S})}\right|=\left|\vvvert
T^{-1/2}({\mathbf{X}}-\overline{\mathbf{X}})\vvvert-\vvvert
T^{-1/2}{\mathbf{X}}\vvvert\right|\leq\vvvert
T^{-1/2}\overline{\mathbf{X}}\vvvert.$ (8)

The last term $\vvvert T^{-1/2}\overline{\mathbf{X}}\vvvert$ is
$\sqrt{\sum_{i=1}^{N}\overline{x}_{i}^{2}}$ (a.s.). To see it, note that the
$it$-entry of $T^{-1}\overline{\mathbf{X}}\,\overline{\mathbf{X}}^{\top}$ is
$T^{-1}\sum_{k=1}^{T}\overline{x}_{i}\overline{x}_{t}=\overline{x}_{i}\overline{x}_{t}$.
Therefore, $[\overline{x}_{1},\ldots,\overline{x}_{N}]^{\top}$ is an
eigenvector of $T^{-1}\overline{\mathbf{X}}\,\overline{\mathbf{X}}^{\top}$,
corresponding to an eigenvalue $\sum_{i=1}^{N}\overline{x}_{i}^{2}$. Clearly,
$\operatorname{rank}\overline{\mathbf{X}}\leq 1$ and almost surely
$\operatorname{rank}\overline{\mathbf{X}}=1$. Thus, it is also the case for
$\operatorname{rank}T^{-1}\overline{\mathbf{X}}\,\overline{\mathbf{X}}^{\top}$.
Considering the eigenvalue $\sum_{i=1}^{N}\overline{x}_{i}^{2}$ of
$T^{-1}\overline{\mathbf{X}}\,\overline{\mathbf{X}}^{\top}$ is positive almost
surely, the other $N-1$ eigenvalues are 0 almost surely.

By dividing (8) with $\sqrt{N}$, it holds almost surely that

$\displaystyle\left|\sqrt{\frac{\lambda_{1}(\mathbf{S}^{\circ})}{N}}-\sqrt{\frac{\lambda_{1}(\mathbf{S})}{N}}\right|\leq\sqrt{\sum_{i=1}^{N}\frac{\overline{x}_{i}^{2}}{N}}\leq\sqrt{\max_{1\leq
i\leq N}\left({\sum_{t=1}^{T}}\frac{x_{it}}{T}\right)^{2}}=\max_{1\leq i\leq
N}\left|{\sum_{t=1}^{T}}\frac{x_{it}}{T}\right|.$ (9)

By $\bm{\Delta}=\mathbf{I}$ and $\bm{\mu}=\bm{0}$, $x_{it}$ $(i,t\geq 1)$ are
standard normal random variables equi-correlated with $\rho\in[0,1)$. By
decomposition (4), there are independent standard normal random variables
${{f}}_{t},e_{it}$ $(i,t\geq 1)$ such that
$x_{it}=\sqrt{\rho}{{f}}_{t}+\sqrt{1-\rho}e_{it}$. By the triangle inequality,

$\displaystyle\max_{1\leq i\leq
N}\left|{\sum_{t=1}^{T}}\frac{x_{it}}{T}\right|\leq\left|{\sum_{t=1}^{T}}\frac{\sqrt{\rho}{{f}}_{t}}{T}\right|+\max_{1\leq
i\leq N}\left|{\sum_{t=1}^{T}}\frac{\sqrt{1-\rho}e_{it}}{T}\right|.$

The first term of the right side tends to 0 in $T\to\infty$ almost surely, by
the law of large numbers. The second term of the right side tends to 0 in
$N,T\to\infty,\ {T/N\to Q}$ almost surely, by Proposition 5.5 with
$\alpha=\beta=1$ and $M=Q$. Thus,

$\displaystyle\max_{1\leq i\leq N}|\overline{x}_{i}|=\max_{1\leq i\leq
N}\left|{\sum_{t=1}^{T}}\frac{x_{it}}{T}\right|{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0\quad({{N,T\to\infty,\ {T/N\to Q}}}).$ (10)

Hence, $\eqref{qq}\mathrel{\stackrel{{\scriptstyle a.s.}}{{\to}}}0$. Since we
have already established
${\lambda_{1}(\mathbf{S})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ for every $\rho\geq 0$, we then derived
${\lambda_{1}(\mathbf{S}^{\circ})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ for every $\rho\geq 0$.

We consider the case ${\mathbf{P}}={\mathbf{\tilde{C}}}$ and
$\bm{\mu}=\bm{0}$. Because the noncentered sample correlation matrix
${\mathbf{\tilde{C}}}$ is invariant under the scaling, we can assume

$\displaystyle\bm{\Delta}=\mathbf{I}.$

From ${\lambda_{1}(\mathbf{S})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ of Theorem 5.2, we will guarantee

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\sqrt{\frac{\lambda_{1}({\mathbf{\tilde{C}}})}{N}}-\sqrt{\frac{\lambda_{1}(\mathbf{S})}{N}}\right|=0\quad(a.s.).$
(11)

For a diagonal matrix

$\displaystyle{{\tilde{\mathbf{D}}}}=\operatorname{diag}\left(\frac{\sqrt{T}}{\lVert{\bm{x}_{1}}\rVert},\ldots,\frac{\sqrt{T}}{\lVert{\bm{x}_{N}}\rVert}\right),\quad\lVert\bm{x}_{i}\rVert=\sqrt{{\sum_{t=1}^{T}}x_{it}^{2}}$

we have ${\mathbf{\tilde{Y}}}=T^{-1/2}{{\tilde{\mathbf{D}}}}{\mathbf{X}}$.

Recall that the _spectral norm_ of a matrix
$\mathbf{M}\in{{\mathbb{R}}}^{m\times n}$ is

$\vvvert\mathbf{M}\vvvert=\sup_{\bm{X}\in{{\mathbb{R}}}^{n},\lVert\bm{X}\rVert=1}\lVert\mathbf{M}\bm{X}\rVert.$

Note
$\vvvert\mathbf{M}\vvvert=\sqrt{\lambda_{1}(\mathbf{M}^{\top}\mathbf{M})}$. By
the triangle inequality of the spectral norm and that
$\vvvert\mathbf{M}_{1}\mathbf{M}_{2}\vvvert\leq\vvvert\mathbf{M}_{1}\vvvert\vvvert\mathbf{M}_{2}\vvvert$
for every $\mathbf{M}_{1}\in{{\mathbb{R}}}^{N\times N}$ and
$\mathbf{M}_{2}\in{{\mathbb{R}}}^{N\times T}$,

$\displaystyle\left|\sqrt{\frac{\lambda_{1}({\mathbf{\tilde{C}}})}{N}}-\sqrt{\frac{\lambda_{1}(\mathbf{S})}{N}}\right|\leq\left\vvvert\frac{1}{\sqrt{N}}{\mathbf{\tilde{Y}}}-\frac{1}{\sqrt{NT}}{\mathbf{X}}\right\vvvert=\vvvert(NT)^{-1/2}({{\tilde{\mathbf{D}}}}-\mathbf{I}){\mathbf{X}}\vvvert$
(12)
$\displaystyle\leq\vvvert{{\tilde{\mathbf{D}}}}-\mathbf{I}\vvvert\cdot\vvvert(NT)^{-1/2}{\mathbf{X}}\vvvert=\max_{{1\leq
i\leq
N}}\left|\frac{T^{1/2}}{\lVert\bm{x}_{i}\rVert}-1\right|\cdot\sqrt{\frac{\lambda_{1}(\mathbf{S})}{N}}.$
(13)

Since we established
${\lambda_{1}(\mathbf{S})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ from $\bm{\Delta}=\mathbf{I}$ and $\bm{\mu}=\bm{0}$, it
suffices to verify

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{{1\leq i\leq
N}}}\left|\frac{\sqrt{T}}{\lVert\bm{x}_{i}\rVert}-1\right|=0\quad(a.s.).$ (14)

By decomposition (4), there are independent standard normal random variables
${{f}}_{t},e_{it}$ $(i,t\geq 1)$ such that
$x_{it}=\sqrt{\rho}{{f}}_{t}+\sqrt{1-\rho}e_{it}$. Hence,

$\displaystyle{\sum_{t=1}^{T}}\frac{x_{it}^{2}}{T}=\rho{\sum_{t=1}^{T}}\frac{{{f}}_{t}^{2}}{T}+2\sqrt{\rho(1-\rho)}{\sum_{t=1}^{T}}\frac{{{f}}_{t}e_{it}}{T}+(1-\rho){\sum_{t=1}^{T}}\frac{e_{it}^{2}}{T}.$

Then

$\displaystyle\max_{{{1\leq i\leq
N}}}\left|\frac{\lVert\bm{x}_{i}\rVert^{2}}{T}-1\right|=\max_{{{1\leq i\leq
N}}}\left|{\sum_{t=1}^{T}}\frac{x_{it}^{2}}{T}-1\right|$ $\displaystyle\leq$
$\displaystyle\rho\max_{{{1\leq i\leq
N}}}\left|{\sum_{t=1}^{T}}\frac{{{f}}_{t}^{2}}{T}-1\right|+2\sqrt{\rho(1-\rho)}\max_{{1\leq
i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{{{f}}_{t}e_{it}}{T}\right|+(1-\rho)\max_{{{1\leq
i\leq N}}}\left|{\sum_{t=1}^{T}}\frac{e_{it}^{2}}{T}-1\right|.$

The right side converges almost surely to 0 by Proposition 5.5, because
${{f}}_{t}^{2}$ $(t\geq 1)$ are i.i.d. of unit mean, ${{f}}_{t}e_{it}$
$(i,t\geq 1)$ are centered i.i.d., and $e_{it}^{2}$ $(i,t\geq 1)$ are i.i.d.
of unit mean. Hence,

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{{1\leq i\leq
N}}}\left|\frac{\lVert\bm{x}_{i}\rVert^{2}}{T}-1\right|=0\quad(a.s.),$ (15)

from which
$\left|{\lVert\bm{x}_{i}\rVert^{2}}/{T}-1\right|>\left|{\lVert\bm{x}_{i}\rVert}/{\sqrt{T}}-1\right|$
implies

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{{1\leq i\leq
N}}}\left|\frac{\lVert\bm{x}_{i}\rVert}{\sqrt{T}}-1\right|=0\quad(a.s.).$ (16)

Suppose $\max_{1\leq i\leq
N}\left|{\lVert\bm{x}_{i}\rVert}/{\sqrt{T}}-1\right|<\epsilon$ for
sufficiently small $\epsilon>0$. Then, for all $i$ $(1\leq i\leq N)$,
$1-\epsilon<{\lVert\bm{x}_{i}\rVert}/\sqrt{T}<1+\epsilon$, which implies
$(1+\epsilon)^{-1}-1<\sqrt{T}/{\lVert\bm{x}_{i}\rVert}-1<(1-\epsilon)^{-1}-1$.
Hence, for all $i$ $(1\leq i\leq N)$,
$\left|\sqrt{T}/{\lVert\bm{x}_{i}\rVert}-1\right|<\max((1-\epsilon)^{-1}-1,\
\left|(1+\epsilon)^{-1}-1\right|)<\epsilon+\epsilon^{2}+\epsilon^{3}+\cdots<2\epsilon$,
because $0<\epsilon\ll 1$. Thus, $\max_{1\leq i\leq
N}\left|\sqrt{T}/{\lVert\bm{x}_{i}\rVert}-1\right|<2\epsilon$. Therefore, from
(16), (14) follows.

Thus, (11) is guaranteed. Therefore,
${\lambda_{1}({\mathbf{\tilde{C}}})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ follows because the assumption $\bm{\mu}=\bm{0}$ implies
${\lambda_{1}(\mathbf{S})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$.

At last, we will demonstrate
${\lambda_{1}(\mathbf{C})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ of Theorem 5.2. Because the sample correlation matrix
$\mathbf{C}$ is invariant under the shifting and scaling, we can assume
$\bm{\Delta}=\mathbf{I}$ and $\bm{\mu}=\bm{0}$. For
$\mathbf{C}={\mathbf{Y}}{\mathbf{Y}}^{\top}$, we have
${\mathbf{Y}}=T^{-1/2}\mathbf{D}({\mathbf{X}}-\overline{\mathbf{X}})$ where

$\displaystyle\mathbf{D}=\operatorname{diag}\left({\sqrt{T}}/{\lVert{\bm{x}_{1}-{\overline{\bm{x}}}_{1}}\rVert},\ldots,{\sqrt{T}}/{\lVert{\bm{x}_{N}-{\overline{\bm{x}}}_{N}}\rVert}\right).$

Recall
$\mathbf{S}^{\circ}=T^{-1}({\mathbf{X}}-\overline{\mathbf{X}})({\mathbf{X}}-\overline{\mathbf{X}})^{\top}$.
By the same arguments as in (13),

$\displaystyle\left|\sqrt{\frac{\lambda_{1}(\mathbf{C})}{N}}-\sqrt{\frac{\lambda_{1}(\mathbf{S}^{\circ})}{N}}\right|\leq\max_{{1\leq
i\leq
N}}\left|\frac{\sqrt{T}}{\lVert{\bm{x}_{i}-{\overline{\bm{x}}}_{i}}\rVert}-1\right|\cdot\sqrt{\frac{\lambda_{1}(\mathbf{S}^{\circ})}{N}}.$

Because ${\lambda_{1}(\mathbf{S}^{\circ})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ follows from $\bm{\Delta}=\mathbf{I}$ we assumed, it
suffices to show

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{1\leq i\leq
N}}\left|\frac{\sqrt{T}}{\lVert{\bm{x}_{i}-{\overline{\bm{x}}}_{i}}\rVert}-1\right|=0\quad(a.s.).$
(17)

By
$\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert^{2}=\lVert\bm{x}_{i}\rVert^{2}-T\left|\overline{x}_{i}\right|^{2}$,

$\displaystyle\max_{{{1\leq i\leq
N}}}\left|\frac{\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert^{2}}{T}-1\right|\leq\max_{{{1\leq
i\leq N}}}\left|\frac{\lVert\bm{x}_{i}\rVert^{2}}{T}-1\right|+\max_{{{1\leq
i\leq N}}}\left|\overline{x}_{i}\right|^{2}.$

The first (second, resp.) term of the right side converges almost surely to 0,
by (15) ((10), resp.). In a similar way as above, (17) is shown. Therefore,
${\lambda_{1}(\mathbf{C})}/{N}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$ for every $\rho\geq 0$. This completes the proof of
Theorem 5.2.

To prove Theorem 1.3 (2), we employ:

######  Proposition 5.6 (Merlevède _et al._ (2019, Theorem 2.2)).

Assume the three assumptions of Proposition 5.4. From a _spectral gap
condition_ on ${{\mathbf{\Sigma}}}$:

$\displaystyle\limsup_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\frac{\lambda_{2}({{\mathbf{\Sigma}}})}{\lambda_{1}({{\mathbf{\Sigma}}})}<1,$
(18)

it follows that

$\displaystyle\sqrt{T}\left(\frac{\lambda_{1}(\mathbf{S})}{\lambda_{1}({{\mathbf{\Sigma}}})}-1-\frac{1}{T}\sum_{k=2}^{N}\frac{\lambda_{k}({{\mathbf{\Sigma}}})}{\lambda_{1}({\mathbf{\Sigma}})-\lambda_{k}({\mathbf{\Sigma}})}\right)\mathrel{\stackrel{{\scriptstyle
D}}{{\to}}}\mathrm{N}(0,\ 2).$

The spectral gap condition (18) holds for
${\mathbf{\Sigma}}={\mathbf{\Sigma}}_{\rho}$ with $\rho>0$, because (3)
implies

$\limsup_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\frac{\lambda_{2}({\mathbf{\Sigma}})}{\lambda_{1}({{\mathbf{\Sigma}}})}=\limsup_{{\begin{subarray}{c}N,T\to\infty\\\
{T/N\to Q}\end{subarray}}}\frac{1-\rho}{(N-1)\rho+1}=0.$

Then,
${T}^{-1}\sum_{k=2}^{N}{\lambda_{k}({{\mathbf{\Sigma}}})}/{\left(\lambda_{1}({{\mathbf{\Sigma}}})-\lambda_{k}({{\mathbf{\Sigma}}})\right)}={{(N-1)(1-\rho)}/{(NT\rho)}}$.
This completes the proof of Theorem 1.3 (2).

##  6 Proof of Corollary 1.4

From Proposition 1.2, $F^{\mathbf{C}}(x)$ converges to
$\mathrm{MP}_{Q,\,1-\rho}(x)$ almost surely for every $x\neq 0$. It suffices
to prove that $\mathrm{MP}_{T/N,\,1-\lambda_{1}(\mathbf{C})/N}(x)$ converges
almost surely to $\mathrm{MP}_{Q,\,1-\rho}(x)$ for any $x\neq 0$. By Theorem
1.3, $\lambda_{1}(\mathbf{C})/N\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\rho$, so
$x/(1-\lambda_{1}(\mathbf{C})/N))\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}x/(1-\rho)>0$. The distribution function $\mathrm{MP}_{Q}(t)$ of
a Marčenko-Pastur distribution is continuous with respect to $t>0$. Thus,

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\mathrm{MP}_{T/N}\left(\frac{x}{1-\lambda_{1}(\mathbf{C})/N}\right)-\mathrm{MP}_{T/N}\left(\frac{x}{1-\rho}\right)\right|=0\quad(a.s.).$
(19)

On the other hand, $\mathrm{MP}_{Q}(t)$ is continuous with respect to
$Q\in(0,\infty)$ for each $t>0$, in view of the density function (1).
Therefore,

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\mathrm{MP}_{T/N}\left(\frac{x}{1-\rho}\right)-\mathrm{MP}_{Q}\left(\frac{x}{1-\rho}\right)\right|=0\quad(a.s).$

By this, (19), and the triangle inequality,

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\mathrm{MP}_{T/N}\left(\frac{x}{1-\lambda_{1}(\mathbf{C})/N}\right)-\mathrm{MP}_{Q}\left(\frac{x}{1-\rho}\right)\right|=0\quad(a.s.).$

##  7 Proof of Theorem 2.3

###  7.1 LSD of sample covariance matrix generated from factor model

In this subsection, we provide the LSD of the sample covariance matrix
$\mathbf{S}$ generated from a factor model (Definition 2.1) in
$N,T\to\infty,{T/N\to Q}$ with Assumption 2. Let $F_{1}$ and $F_{2}$ be
distribution functions. The _Lévy distance_ between $F_{1}$ and $F_{2}$ is
denoted by $\operatorname{d_{L}}(F_{1},F_{2})$. For the definition, see (Huber
& Ronchetti, 2009, Definition 2.7). By (Huber & Ronchetti, 2009, Theorem 2.9),

######  Proposition 7.1.

For every sequence $(F_{n})_{n}$ of distribution functions, and every
distribution function $F$, $F_{n}$ converges weakly to $F$ if and only if
$\operatorname{d_{L}}(F_{n},F)\to 0$.

By Huber & Ronchetti (2009, p. 36), the _Kolmogorov distance_ between two
$F_{1}$ and $F_{2}$ is defined as
$\operatorname{d_{K}}(F_{1},F_{2})=\sup_{x\in{{\mathbb{R}}}}\left|F_{1}(x)-F_{2}(x)\right|$,
and satisfies

$\displaystyle\operatorname{d_{L}}(F_{1},F_{2})\leq\operatorname{d_{K}}(F_{1},F_{2}).$
(20)

######  Proposition 7.2 (Bai (1999, Lemma 2.6, the rank inequality)).

$\displaystyle\operatorname{d_{K}}\left(F^{{\mathbf{A}}{\mathbf{A}}^{\top}},F^{\mathbf{B}\mathbf{B}^{\top}}\right)\leq\frac{1}{N}\;\operatorname{rank}({\mathbf{A}}-\mathbf{B}),\qquad\left({\mathbf{A}},\mathbf{B}\in{{\mathbb{R}}}^{N\times
T}\right).$

######  Theorem 7.3 (Scaling).

Given a factor model. Suppose ${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$
and Assumption 2. Then, the following two hold almost surely:

  1. 1.

$F^{\mathbf{S}}$ converges weakly almost surely to
$\mathrm{MP}_{Q,\sigma_{\infty}^{2}(1-\rho)}$, if $\mu_{i}=0$ $(1\leq i\leq
N)$.

  2. 2.

$F^{\mathbf{E}\mathbf{E}^{\top}}$ converges weakly almost surely to
$\mathrm{MP}_{Q,\sigma_{\infty}^{2}(1-\rho)}$.

######  Proof 7.4.

By Definition 2.1, the data matrix is written as

$\displaystyle{\mathbf{X}}={\sum_{k=1}^{K}}\left[\ell_{ik}f_{tk}\right]_{it}+\bm{\Psi},\quad\bm{\Psi}:=[e_{it}]_{it}\in{{\mathbb{R}}}^{N\times
T}.$ (21)

By Assumption 2 and Proposition 1.1 (1), the ESD of
$T^{-1}\bm{\Psi}\bm{\Psi}^{\top}$ converges weakly to
$\mathrm{MP}_{Q,\psi_{1}}$ almost surely. Then, Proposition 7.1 implies

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\operatorname{d_{L}}\left(F^{{T}^{-1}\bm{\Psi}\bm{\Psi}^{\top}},\,\mathrm{MP}_{Q,\psi_{1}}\right)=0\quad(a.s.)$
(22)

where $\psi_{1}=\sigma_{\infty}^{2}(1-\rho)$ by Assumption 2. Recall
$\mathbf{S}=T^{-1}{\mathbf{X}}{\mathbf{X}}^{\top}$. Thus, by (20), Proposition
7.2, (21), and $\operatorname{rank}[\ell_{ik}f_{tk}]_{it}\leq 1$ $(1\leq k\leq
K)$, it holds almost surely that

$\displaystyle\operatorname{d_{L}}\left(F^{\mathbf{S}},\
F^{{T}^{-1}\bm{\Psi}\bm{\Psi}^{\top}}\right)\leq\operatorname{d_{K}}\left(F^{\mathbf{S}},\
F^{{T}^{-1}\bm{\Psi}\bm{\Psi}^{\top}}\right)\leq\frac{1}{N}\operatorname{rank}\left(\frac{1}{\sqrt{T}}{\mathbf{X}}-\frac{1}{\sqrt{T}}\bm{\Psi}\right)$
$\displaystyle=\frac{1}{N}\operatorname{rank}\left(\frac{1}{\sqrt{T}}{\sum_{k=1}^{K}}\left[\ell_{ik}f_{tk}\right]_{it}\right)\leq\frac{1}{N}{\sum_{k=1}^{K}}\operatorname{rank}\left(\left[\ell_{ik}f_{tk}\right]_{it}\right)\leq\frac{K}{N}\to
0\qquad(N\to\infty).$

By this, (22), and the triangle inequality,

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\operatorname{d_{L}}\left(F^{\mathbf{S}},\
\mathrm{MP}_{Q,\psi_{1}}\right)=0\quad(a.s.),$ (23)

which implies the first assertion of this theorem through Proposition 7.1. By
the same proposition, the second assertion of this theorem follows by the
triangle inequality from (23) and

$\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\operatorname{d_{L}}\left(F^{\mathbf{E}\mathbf{E}^{\top}},\
\mathrm{MP}_{Q,\psi_{1}}\right)=0\quad(a.s.).$

The last is because by (20), Proposition 7.2, and
$\operatorname{rank}(\overline{\mathbf{X}})\leq 1$, it holds almost surely
that

$\displaystyle\operatorname{d_{L}}(F^{\mathbf{E}\mathbf{E}^{\top}},\
F^{\mathbf{S}})\leq\operatorname{d_{K}}\left(F^{\mathbf{E}\mathbf{E}^{\top}},\
F^{{T}^{-1}{{\mathbf{X}}{\mathbf{X}}^{\top}}}\right)\leq\frac{\operatorname{rank}\left(\overline{\mathbf{X}}\right)}{N}\leq{1}/{N}\to
0\quad(N\to\infty).$

This completes the proof of Theorem 7.3.

###  7.2 LSD of sample correlation matrix generated from factor model

In this subsection, we provide the LSD of $\mathbf{C}$ generated from a factor
model (Definition 2.1) in $N,T\to\infty,{T/N\to Q}$ with Assumption 2. By
Assumption 2,

$\displaystyle L:=\sup\set{\left|\ell_{ik}\right|\colon{1\leq i,\ 1\leq k\leq
K}}<\infty.$ (24)

For $\mu_{i}=0$ $(1\leq i\leq N)$,

$\displaystyle{\sum_{t=1}^{T}}\frac{x_{it}^{2}}{T}=\frac{\operatorname{Tr}\left(\bm{\ell}_{i}^{\top}{\bm{\ell}}_{i}{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}\right)}{T}+2{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it}}{T}+{\sum_{t=1}^{T}}\frac{e_{it}^{2}}{T}$
(25)

because

$\frac{1}{T}{\sum_{t=1}^{T}}{{\bm{f}}}_{t}^{\top}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}{{\bm{f}}}_{t}=\operatorname{Tr}\left({\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}\right).$

######  Lemma 7.5.

For a factor model with $\mu_{i}=0$ $(1\leq i\leq N)$, if Assumption 2, then

$\displaystyle\lim_{{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}}\frac{1}{N}\operatorname{Tr}\mathbf{S}=\sigma_{\infty}^{2}\quad(a.s.).$

######  Proof 7.6.

By (25),
$\operatorname{Tr}\mathbf{S}/N=\sum_{i=1}^{N}{\sum_{t=1}^{T}}{x_{it}^{2}}/{(NT)}$
is equal to

$\displaystyle\operatorname{Tr}\left(\frac{\sum_{i=1}^{N}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}}{N}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}\right)+2\frac{\sum_{i=1}^{N}{\sum_{t=1}^{T}}{\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it}}{NT}+\frac{\sum_{i=1}^{N}{\sum_{t=1}^{T}}e_{it}^{2}}{NT}.$
(26)

We will compute the almost sure limits of the three terms of (26), in
$N,T\to\infty,{T/N\to Q}$.

By the continuous mapping theorem (Van der Vaart, 2000, Theorem 2.3) applied
to the continuous function $\operatorname{Tr}$, the limit of the first term of
(26) is

$\displaystyle\operatorname{Tr}\left(\lim_{{{\begin{subarray}{c}N,T\to\infty\\\
{T/N\to
Q}\end{subarray}}}}\frac{\sum_{i=1}^{N}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}}{N}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}\right)=\operatorname{Tr}\left(\lim_{N\to\infty}\frac{\sum_{i=1}^{N}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}}{N}\lim_{T\to\infty}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}\right).$
(27)

By Assumption 2,
$\lim_{i\to\infty}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}=\bm{\ell}^{\top}\bm{\ell}$,
so the Cesàro sum (Abbott, 2015) satisfies
$\lim_{N\to\infty}{\sum_{i=1}^{N}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}}/{N}=\bm{\ell}^{\top}\bm{\ell}.$
On other hand, by Definition 2.1, $f_{tk}$ $(1\leq t\leq T,\ 1\leq k\leq K)$
are centered i.i.d. Thus, $f_{tk}f_{tl}$ $(t=1,2,\ldots)$ are i.i.d., for each
$k,l\in\\{1,\ldots,K\\}$. Since $\operatorname{E}(f_{tk}f_{tl})=1\ (k=l);0\
(k\neq l)$, the strong law of large numbers implies that

$\displaystyle\lim_{T\to\infty}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}=\operatorname{E}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}=\mathbf{I}_{K}\quad(a.s.).$
(28)

By this, the limit of the first term of (26) is:

$\displaystyle\lim_{{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}}\operatorname{Tr}\left(\frac{\sum_{i=1}^{N}{\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}}{N}\frac{{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}}{T}\right)=\operatorname{Tr}\left(\bm{\ell}^{\top}\bm{\ell}\mathbf{I}_{K}\right)=\lVert\bm{\ell}\rVert^{2}\quad(a.s.).$
(29)

Next, we consider the second term of (26). We will verify:

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}2\frac{\sum_{i=1}^{N}{\sum_{t=1}^{T}}{\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it}}{NT}=0\quad(a.s.).$
(30)

Let $(i_{j},\ t_{j})$ $(j=1,2,3,\ldots)$ be an enumeration of
$\set{(i,t)\colon\,i,t\in\mathbb{Z}_{\geq 1}}$. Because $f_{t_{j},k}$ $(j\geq
1,\ 1\leq k\leq K)$ and $e_{i_{j},t_{j}}$ $(j\geq 1)$ are independent and
centered, so are ${\bm{\ell}}_{i_{j}}{{\bm{f}}}_{t_{j}}$ and
$e_{i_{j},t_{j}}$. Then,
$\sum_{j=1}^{\infty}j^{-2}\operatorname{Var}\left({\bm{\ell}}_{i_{j}}{{\bm{f}}}_{t_{j}}e_{i_{j},t_{j}}\right)=\sum_{j=1}^{\infty}j^{-2}\operatorname{Var}\left({\bm{\ell}}_{i_{j}}{{\bm{f}}}_{t_{j}}\right)\operatorname{Var}e_{i_{j},t_{j}}=\sum_{j=1}^{\infty}j^{-2}\operatorname{Var}\left({\bm{\ell}}_{i_{j}}{{\bm{f}}}_{t_{j}}\right)\psi_{1}$.
Because the $K$ entries of
${{\bm{f}}}_{t_{j}}=[f_{t_{j}1},\ldots,f_{t_{j}K}]^{\top}$ are independent
random variables with unit variance for each $j$, (24) implies
$\sum_{j=1}^{\infty}{\operatorname{Var}\left({\bm{\ell}}_{i_{j}}{{\bm{f}}}_{t_{j}}\right)}/{j^{2}}=\sum_{j=1}^{\infty}{\lVert{\bm{\ell}}_{i_{j}}\rVert^{2}}/{j^{2}}<\infty$.
Moreover,
$\operatorname{E}({\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it})=\operatorname{E}({\bm{\ell}}_{i}{{\bm{f}}}_{t})\operatorname{E}(e_{it})=0$.
Hence, (30) follows from:

######  Proposition 7.7 (Kolmogorov sufficient condition (Gut, 2013, Theorem
6.5.4)).

Let $X_{1},X_{2},\ldots$ be centered, independent random variables with finite
variance. Then $\sum_{j=1}^{\infty}{\operatorname{Var}X_{j}}/{j^{2}}<\infty$
implies $\lim_{n\to\infty}{\sum_{j=1}^{n}X_{j}}/{n}=0$ (a.s.).

The third term of (26) tends almost surely to $\psi_{1}$, by the strong law of
large numbers. Thus, by (29) and (30), we get
$\eqref{lem:weak}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}\lVert\bm{\ell}\rVert^{2}+\psi_{1}=\sigma_{\infty}^{2}$. This
completes the proof of Lemma 7.5.

Now, we prove Theorem 2.3.

######  Proof 7.8.

The sample correlation matrix $\mathbf{C}$ and the noncentered sample
correlation matrix ${\mathbf{\tilde{C}}}$ (Definition 5.1) are invariant under
scaling of variables. To the factor model, we can assume

$\displaystyle\ell_{it}:=\frac{\ell_{it}}{\sigma_{\infty}},\quad\psi_{1}:=\frac{\psi_{1}}{\sigma_{\infty}^{2}}.$
(31)

without loss of generality. Moreover, we can safely assume that $\mu_{i}=0$
$(1\leq i\leq N)$ to prove that $F^{\mathbf{C}}$ converges weakly to
$\mathrm{MP}_{Q,1-\rho}$ almost surely, because $\mathbf{C}$ is invariant
under shifting. It suffices to show that
$\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\operatorname{d_{L}}\left(F^{\mathbf{C}},\
F^{\mathbf{E}\mathbf{E}^{\top}}\right)=0$ (a.s.), because Theorem 7.3 (2)
implies $\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\operatorname{d_{L}}\left(F^{\mathbf{E}\mathbf{E}^{\top}},\,\mathrm{MP}_{Q,1-\rho}\right)=0$
(a.s.). By $\mathbf{C}={\mathbf{Y}}{\mathbf{Y}}^{\top}$, (Bai, 1999, Lemma
2.7) gives an upper bound on the fourth power of the Lévy distance

$\displaystyle\operatorname{d_{L}}^{4}\left(F^{\mathbf{C}},\
F^{\mathbf{E}\mathbf{E}^{\top}}\right)\leq\frac{2}{N}\operatorname{Tr}\left({\mathbf{Y}}{\mathbf{Y}}^{\top}+\mathbf{E}\mathbf{E}^{\top}\right)\times\frac{1}{N}\operatorname{Tr}\left(\left({\mathbf{Y}}-\mathbf{E}\right)\left({\mathbf{Y}}-\mathbf{E}\right)^{\top}\right).$
(32)

On the right side of (32),
$\operatorname{Tr}({\mathbf{Y}}{\mathbf{Y}}^{\top})/N=\operatorname{Tr}\mathbf{C}/N=1$,
and

$\displaystyle\frac{\operatorname{Tr}(\mathbf{E}\mathbf{E}^{\top})}{N}$
$\displaystyle=\sum_{i=1}^{N}{\sum_{t=1}^{T}}\frac{\left(x_{it}-\overline{x}_{i}\right)^{2}}{NT}=\sum_{i=1}^{N}{\sum_{t=1}^{T}}\frac{x_{it}^{2}}{NT}-\sum_{i=1}^{N}\frac{\overline{x}_{i}^{2}}{N}\leq\sum_{i=1}^{N}{\sum_{t=1}^{T}}\frac{x_{it}^{2}}{NT}.$
(33)

Since the rightmost term of (33) converges almost surely to a finite
deterministic value by Lemma 7.5,
$\limsup_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\left|\operatorname{Tr}(\mathbf{E}\mathbf{E}^{\top})/N\right|<\infty$
(a.s.). Therefore, for the right side of (32), we have only to assure the
following:

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\frac{1}{N}\operatorname{Tr}\left(\left({\mathbf{Y}}-\mathbf{E}\right)\left({\mathbf{Y}}-\mathbf{E}\right)^{\top}\right)=0\quad(a.s.).$
(34)

On the other hand, by ${\mathbf{\tilde{C}}}=T^{-1}{\mathbf{X}}{\mathbf{X}}$
and (Bai, 1999, Lemma 2.7),
$\operatorname{d_{L}}^{4}\left(F^{\mathbf{\tilde{C}}},\
F^{T^{-1}{\mathbf{X}}{\mathbf{X}}^{\top}}\right)$ is at most the product of
${2}N^{-1}\operatorname{Tr}\left({\mathbf{\tilde{Y}}}{\mathbf{\tilde{Y}}}^{\top}+T^{-1}{\mathbf{X}}{\mathbf{X}}^{\top}\right)$
and
$N^{-1}\operatorname{Tr}\left(\left({\mathbf{\tilde{Y}}}-T^{-1/2}{\mathbf{X}}\right)\left({\mathbf{\tilde{Y}}}-T^{-1/2}{\mathbf{X}}\right)^{\top}\right)$.
Here,
$\operatorname{Tr}{\mathbf{\tilde{Y}}}{\mathbf{\tilde{Y}}}^{\top}/N=\operatorname{Tr}{\mathbf{\tilde{C}}}/N=1$
and
$\left|\operatorname{Tr}\left((NT)^{-1}{\mathbf{X}}{\mathbf{X}}^{\top}\right)\right|<\infty$
(a.s.) by Lemma 7.5. Hence, to prove that $F^{{\mathbf{\tilde{C}}}}$ converges
weakly to $\mathrm{MP}_{Q,1-\rho}$ almost surely, it suffices to guarantee

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\frac{1}{N}\operatorname{Tr}\left(\left({\mathbf{\tilde{Y}}}-T^{-1/2}{\mathbf{X}}\right)\left({\mathbf{\tilde{Y}}}-T^{-1/2}{\mathbf{X}}\right)^{\top}\right)=0\quad(a.s.),$
(35)

We will prove (35), and then derive (34) from it.

(a) The proof of (35). The left side of (35) is
${\tilde{d}}_{1}-2{\tilde{d}}_{2}$ where

$\displaystyle{\tilde{d}}_{1}=\frac{1}{NT}\sum_{i=1}^{N}{\sum_{t=1}^{T}}{x_{it}^{2}}-1,\quad{\tilde{d}}_{2}=\frac{1}{N}\sum_{i=1}^{N}\sqrt{\frac{1}{T}{\sum_{t=1}^{T}}x_{it}^{2}}-1,$
(36)

because
$\operatorname{Tr}({\mathbf{\tilde{Y}}}{\mathbf{\tilde{Y}}}^{\top})=\operatorname{Tr}({\mathbf{\tilde{C}}})=N$
by the definition of ${\mathbf{\tilde{C}}}$ and because
$\operatorname{Tr}({\mathbf{X}}{\mathbf{\tilde{Y}}}^{\top})=\operatorname{Tr}\left(\left[{\mathbf{X}}{\bm{x}_{1}^{\top}}/{\lVert\bm{x}_{1}\rVert},\ldots,{\mathbf{X}}{\bm{x}^{\top}_{N}}/{\lVert\bm{x}_{N}\rVert}\right]\right)=\sum_{i=1}^{N}{\bm{x}_{i}\bm{x}_{i}^{\top}}/{\lVert\bm{x}_{i}\rVert}=\sum_{i=1}^{N}\lVert\bm{x}_{i}\rVert.$
Since Lemma 7.5 implies

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}{\tilde{d}}_{1}=0\quad(a.s.),$ (37)

our goal (35) follows from $\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}{\tilde{d}}_{2}=0$ (a.s.). Because of this, we will verify

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\frac{1}{T}{\sum_{t=1}^{T}}x_{it}^{2}=1\quad(a.s.).$ (38)

By (25), ${\sum_{t=1}^{T}}{x_{it}^{2}}/T$ is

$\displaystyle\frac{\operatorname{Tr}\left({\bm{\ell}}_{i}^{\top}{\bm{\ell}}_{i}{\sum_{t=1}^{T}}{{\bm{f}}}_{t}{{\bm{f}}}_{t}^{\top}\right)}{T}+2{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it}}{T}+{\sum_{t=1}^{T}}\frac{e_{it}^{2}-\psi_{1}}{T}+\psi_{1}.$
(39)

The first term of (39) tends almost surely to $\lVert\bm{\ell}_{i}\rVert^{2}$
in ${{N,T\to\infty,\ {T/N\to Q}}}$, by (28) and the continuous mapping theorem
for the continuous function $\operatorname{Tr}$.

Next, we will verify that the second term of (39) converges almost surely to
$0$, similarly as (30). Because $f_{tk}$ $(t\geq 1,\ 1\leq k\leq K)$ and
$e_{it}$ $(t\geq 1)$ are independent and centered, so are
${\bm{\ell}}_{i}{{\bm{f}}}_{t}$ and $e_{it}$. By $\operatorname{Var}f_{tk}=1$
$(1\leq k\leq K)$, we then have
$\sum_{t=1}^{\infty}t^{-2}\operatorname{Var}\left({\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{i,t}\right)=\sum_{t=1}^{\infty}t^{-2}\operatorname{Var}\left({\bm{\ell}}_{i}{{\bm{f}}}_{t}\right)\operatorname{Var}e_{i,t}=\sum_{t=1}^{\infty}t^{-2}\lVert\bm{\ell}_{i}\rVert^{2}\psi_{1}<\infty$
by (24). Moreover, $\bm{\ell}_{i}\bm{f}_{t}e_{it}$ $(t=1,2,3,\ldots)$ are
centered independent. Hence, by Proposition 7.7, the second term of (39)
converges almost surely to 0:

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}2{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it}}{T}=2\operatorname{E}({\bm{\ell}}_{i}{{\bm{f}}}_{t}e_{it})=2\operatorname{E}({\bm{\ell}}_{i}{{\bm{f}}}_{t})\operatorname{E}(e_{it})=0\quad(a.s.).$

For the third term of (39), we will verify

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{1\leq i\leq
N}}\left|T^{-1}{\sum_{t=1}^{T}}(e_{it}^{2}-\psi_{1})\right|=0\quad(a.s.).$
(40)

Because we suppose ${{N,T\to\infty,\ {T/N\to Q}}}\in(0,\,\infty)$, there
exists $T_{1}(m)$ such that for all $T$,

$\displaystyle T>T_{1}(m)\implies(Q-1/m)N<T<(Q+1/m)N.$ (41)

Since $e_{it}^{2}$ $(1\leq i\leq N,\ 1)$ are i.i.d. and
$\operatorname{E}(e_{11}^{2})=\psi_{1}$, Proposition 5.5 implies: for every
constant $M>0$, almost surely, there exists $T_{2}(m,M)$ such that for all
$T\geq T_{2}(m,M)$ we have $\left|\max_{1\leq i\leq
MT}{\sum_{t=1}^{T}}(e_{it}^{2}-\psi_{1})/T\right|<m^{-1}$. Hence, by (41), for
all $T\geq\max(T_{1}(m),\,T_{2}(m,\,(Q-1/m)^{-1}))$, it holds

$\displaystyle\max_{{1\leq
i\leq\frac{T}{Q+1/m}}}\left|{\sum_{t=1}^{T}}\frac{e_{it}^{2}-\psi_{1}}{T}\right|\leq\max_{{1\leq
i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{e_{it}^{2}-\psi_{1}}{T}\right|\leq\max_{{1\leq
i\leq\frac{T}{Q-1/m}}}\left|{\sum_{t=1}^{T}}\frac{e_{it}^{2}-\psi_{1}}{T}\right|<\frac{1}{m}.$

Consequently, (40) has verified.

To sum up, by (31) and $\psi=1$. Hence, (38) is verified. Thus

$\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}{\tilde{d}}_{2}=0\quad(a.s.).$ (42)

Therefore (35) is concluded.

(b) The proof of (34). First, the left side of (34) is $d_{1}-2d_{2}$ where

$\displaystyle
d_{1}=\frac{1}{NT}\sum_{i=1}^{N}{\sum_{t=1}^{T}}{\left(x_{it}-\overline{x}_{i}\right)^{2}}-1,\quad
d_{2}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert}{\sqrt{T}}-1\right).$
(43)

It is proved in a similar way for (a) as follows: By the definition,
$\operatorname{Tr}({\mathbf{Y}}{\mathbf{Y}}^{\top})=N$. By Definition 5.1, the
$i$-th column of ${\mathbf{Y}}^{\top}$ is
${(\bm{x}_{i}-{\overline{\bm{x}}}_{i})^{\top}}/\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert$,
so that of $\mathbf{E}{\mathbf{Y}}^{\top}$ is
$T^{-1/2}({\mathbf{X}}-\overline{\mathbf{X}})\
{(\bm{x}_{i}-{\overline{\bm{x}}}_{i})^{\top}}/\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert$.
Thus,
$\operatorname{Tr}\left(\mathbf{E}{\mathbf{Y}}^{\top}\right)=\sum_{i=1}^{N}{(\bm{x}_{i}-{\overline{\bm{x}}}_{i})(\bm{x}_{i}-{\overline{\bm{x}}}_{i})^{\top}}/\left(\sqrt{T}\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert\right)=\sum_{i=1}^{N}{\lVert\bm{x}_{i}-{\overline{\bm{x}}}_{i}\rVert}/{\sqrt{T}}.$

Secondly, we will show $d_{1}\mathrel{\stackrel{{\scriptstyle a.s.}}{{\to}}}0$
and $d_{2}\mathrel{\stackrel{{\scriptstyle a.s.}}{{\to}}}0$. From (36) and
(43), it follows immediately

$\displaystyle\left|d_{1}-{\tilde{d}}_{1}\right|$
$\displaystyle=\frac{1}{NT}\left|\sum_{i=1}^{N}{\sum_{t=1}^{T}}{x_{it}^{2}}-\sum_{i=1}^{N}{\sum_{t=1}^{T}}{\left(x_{it}-\overline{x}_{i}\right)^{2}}\right|=\frac{1}{N}\sum_{i=1}^{N}{\left|\overline{x}_{i}\right|^{2}}$
(44)

Thus,

$\displaystyle\left|d_{1}-{\tilde{d}}_{1}\right|\leq\left(\max_{{1\leq i\leq
N}}{\left|\overline{x}_{i}\right|}\right)^{2}\leq\left(\max_{{1\leq i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}}{T}\right|+\max_{{1\leq
i\leq N}}\left|{\sum_{t=1}^{T}}\frac{e_{it}}{T}\right|\right)^{2}.$ (45)

As for the first term inside the right side,

$\displaystyle\max_{{1\leq i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}}{T}\right|\leq\max_{{1\leq
i\leq
N}}\left({\sum_{k=1}^{K}}\left|\ell_{ik}\right|\left|{\sum_{t=1}^{T}}{f_{tk}}/{T}\right|\right).$

By (24),

$\displaystyle\max_{{1\leq i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}}{T}\right|\leq
KL\max_{1\leq k\leq
K}\left|{\sum_{t=1}^{T}}\frac{f_{tk}}{T}\right|\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0\quad\left({{N,T\to\infty,\ {T/N\to Q}}}\right),$ (46)

by the strong law of large numbers, because for each $k$ $(1\leq k\leq K)$,
$f_{tk}=0$ $(t\geq 1)$ are centered i.i.d.

Next, we consider the second term $\max_{{1\leq i\leq
N}}\left|{\sum_{t=1}^{T}}e_{it}/{T}\right|$ inside the right side of (45). By
Proposition 5.5, for every constant $M>0$, almost surely, there exists
$T_{3}(m,M)$ such that for all $T\geq T_{3}(m,M)$ we have $\left|\max_{1\leq
i\leq MT}{\sum_{t=1}^{T}}e_{it}/T\right|<m^{-1}$. Hence, by (41), for all
$T\geq\max(T_{1}(m),\,T_{3}(m,\,(Q-1/m)^{-1}))$, it holds

$\displaystyle\max_{{1\leq
i\leq\frac{T}{Q+1/m}}}\left|{\sum_{t=1}^{T}}\frac{e_{it}}{T}\right|\leq\max_{{1\leq
i\leq N}}\left|{\sum_{t=1}^{T}}\frac{e_{it}}{T}\right|\leq\max_{{1\leq
i\leq\frac{T}{Q-1/m}}}\left|{\sum_{t=1}^{T}}\frac{e_{it}}{T}\right|<\frac{1}{m}.$

Consequently, $\displaystyle\lim_{{\begin{subarray}{c}N,T\to\infty\\\ {T/N\to
Q}\end{subarray}}}\max_{{1\leq i\leq
N}}\left|T^{-1}{\sum_{t=1}^{T}}e_{it}\right|=0$ (a.s.). By this and (46),

$\displaystyle\left|d_{1}-{\tilde{d}}_{1}\right|\leq\left(\max_{{1\leq i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{{\bm{\ell}}_{i}{{\bm{f}}}_{t}}{T}\right|+\max_{{1\leq
i\leq
N}}\left|{\sum_{t=1}^{T}}\frac{e_{it}}{T}\right|\right)^{2}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0\quad({{N,T\to\infty,\ {T/N\to Q}}}).$ (47)

Thus, $d_{1}\mathrel{\stackrel{{\scriptstyle a.s.}}{{\to}}}0$, by (37).

In contrast, by (38) and (43),

$\displaystyle\left|d_{2}-{\tilde{d}}_{2}\right|$
$\displaystyle\leq\frac{1}{N\sqrt{T}}\sum_{i=1}^{N}\left|\sqrt{{\sum_{t=1}^{T}}(x_{it}-\overline{x}_{i})^{2}}-\sqrt{{\sum_{t=1}^{T}}x_{it}^{2}}\right|.$

Note $|\sqrt{r}-\sqrt{s}|\leq|r-s|/\sqrt{s}$ $(r,s\geq 0)$. Let
$r={T}^{-1}{\sum_{t=1}^{T}}(x_{it}-\overline{x}_{i})^{2}$ and
$s={T}^{-1}{\sum_{t=1}^{T}}x_{it}^{2}$. Then, $|r-s|=(\overline{x}_{i})^{2}$.

$\displaystyle\left|d_{2}-{\tilde{d}}_{2}\right|\leq
N^{-1}\sum_{i=1}^{N}\left|\overline{x}_{i}\right|^{2}\cdot\left({\sum_{t=1}^{T}}{x_{it}^{2}}/{T}\right)^{-1/2}.$

On the right side, $N^{-1}\sum_{i=1}^{N}\left|\overline{x}_{i}\right|^{2}$ is
$\left|d_{1}-{\tilde{d}}_{1}\right|$ by (44), which converges almost surely to
0 by (47). Thus, (38) implies
$\left|d_{2}-{\tilde{d}}_{2}\right|\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0$, from which $d_{2}\mathrel{\stackrel{{\scriptstyle
a.s.}}{{\to}}}0$ follows by (42). Therefore, we have demonstrated (34). This
completes the proof of Theorem 2.3.

## Acknowledgments

Kazuyoshi Yata and Makoto Aoshima kindly drew the author’s attention to Yata &
Aoshima (2009, 2010, 2012) and Ishii _et al._ (2021). The author thanks the
reviewers.

## References

  * Abbott (2015) S. Abbott (2015) _Understanding Analysis_ , 2nd edition.  New York, NY: Springer.
  * Akama & Husnaqilati (2022) Y. Akama & A. Husnaqilati (2022) A dichotomous behavior of Guttman-Kaiser criterion from equi-correlated normal population, _J. Indones. Math. Soc._ 28 (3), 272–303. See also arXiv:2210.12580.
  * Bai (2003) J. Bai (2003) Inferential theory for factor models of large dimensions, _Econometrica_ 71 (1), 135–171.
  * Bai (1999) Z. D. Bai (1999) Methodologies in spectral analysis of large dimensional random matrices, a review, _Stat. Sin._ 9 (3), 611–662.
  * Bai & Silverstein (2010) Z. D. Bai & J. W. Silverstein (2010) _Spectral analysis of large dimensional random matrices_ , 2nd edition.  New York, NY: Springer.
  * Bai & Yin (1993) Z. D. Bai & Y. Q. Yin (1993) Limit of the smallest eigenvalue of a large dimensional sample covariance matrix, _Ann Probab_ 21 (3), 1275–1294.
  * Baik _et al._ (2005) J. Baik, G. Ben Arous & S. Péché (2005) Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, _Ann Probab_ 33 (5), 1643–1697.
  * Bun _et al._ (2017) J. Bun, J.-P. Bouchaud & M. Potters (2017) Cleaning large correlation matrices: Tools from random matrix theory, _Phys. Rep._ 666, 1–109.
  * Cai _et al._ (2017) T. Cai, X. Han & G. Pan (2017) Limiting laws for divergent spiked eigenvalues and largest non-spiked eigenvalue of sample covariance matrices, arXiv:1711.00217.
  * Candila (2021) V. Candila (2021) _dccmidas: A package for estimating DCC-based models in R_ , R package version 0.1.0.
  * Donoho _et al._ (2018) D. Donoho, M. Gavish & I. Johnstone (2018) Optimal shrinkage of eigenvalues in the spiked covariance model, _Ann Stat_ 46 (4), 1742–1778.
  * El Karoui (2008) N. El Karoui (2008) Operator norm consistent estimation of large-dimensional sparse covariance matrices, _Ann Stat_ 36 (6), 2717–2756.
  * Elton _et al._ (2014) E. J. Elton, M. J. Gruber, S. J. Brown & W. N. Goetzmann (2014) _Modern portfolio theory and investment analysis_ , 9th edition.  New York, NY: John Wiley & Sons.
  * Engle & Kelly (2012) R. Engle & B. Kelly (2012) Dynamic equicorrelation, _J Bus Econ Stat_ 30 (2), 212–228.
  * Fama & French (1993) E. F. Fama & K. R. French (1993) Common risk factors in the returns on stocks and bonds, _J. Financ. Econ._ 33 (1), 3–56.
  * Fama & French (2015) E. F. Fama & K. R. French (2015) A five-factor asset pricing model, _J. Financ. Econ._ 116 (1), 1–22.
  * Fan & Jiang (2019) J. Fan & T. Jiang (2019) Largest entries of sample correlation matrices from equi-correlated normal populations, _Ann Probab_ 47 (5), 3321–3374.
  * Friedman & Weisberg (1981) S. Friedman & H. F. Weisberg (1981) Interpreting the first eigenvalue of a correlation matrix, _Educ. Psychol. Meas._ 41 (1), 11–21.
  * Glosten _et al._ (1993) L. R. Glosten, R. Jagannathan & D. E. Runkle (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks, _J Finance_ 48 (5), 1779–1801.
  * Gut (2013) A. Gut (2013) _Probability: A Graduate Course_.  New York, NY: Springer.
  * Huber & Ronchetti (2009) P. J. Huber & E. Ronchetti (2009) _Robust statistics_ , 2nd edition.  Hoboken, NJ: John Wiley & Sons.
  * Ishii _et al._ (2021) A. Ishii, K. Yata & M. Aoshima (2021) Hypothesis tests for high-dimensional covariance structures, _Annals of the Institute of Statistical Mathematics_ 73 (3), 599–622.
  * Jiang (2004) T. Jiang (2004) The limiting distributions of eigenvalues of sample correlation matrices, _Sankhyā: The Indian Journal of Statistics (2003–2007)_ 66 (1), 35–48.
  * Johnstone (2001) I. M. Johnstone (2001) On the distribution of the largest eigenvalue in principal components analysis, _Ann Stat_ 29 (2), 295–327.
  * Laloux _et al._ (2000) L. Laloux, P. Cizeau, M. Potters & J.-P. Bouchaud (2000) Random matrix theory and financial correlations, _Int. J. Theor. Appl. Finance_ 3 (3), 391–397.
  * Ledoit & Péché (2011) O. Ledoit & S. Péché (2011) Eigenvectors of some large sample covariance matrix ensembles, _Probab. Theory Relat. Fields_ 151 (1), 233–264.
  * Ledoit & Wolf (2004) O. Ledoit & M. Wolf (2004) Honey, I shrunk the sample covariance matrix, _J. Portf. Manag._ 30 (4), 110–119.
  * Ledoit & Wolf (2012) O. Ledoit & M. Wolf (2012) Nonlinear shrinkage estimation of large-dimensional covariance matrices, _Ann Stat_ 40 (2), 1024–1060.
  * Ledoit & Wolf (2018) O. Ledoit & M. Wolf (2018) Optimal estimation of a large-dimensional covariance matrix under Stein’s loss, _Bernoulli_ 24 (4B), 3791–3832.
  * Markowitz (1959) H. M. Markowitz (1959) _Portfolio selection: efficient diversification of investments_.  New York, NY: John Wiley & Sons.
  * Marčenko & Pastur (1967) V. A. Marčenko & L. A. Pastur (1967) Distribution of eigenvalues in certain sets of random matrices, _Mat. Sb._ 72, 507–536.
  * Merlevède _et al._ (2019) F. Merlevède, J. Najim & P. Tian (2019) Unbounded largest eigenvalue of large sample covariance matrices: Asymptotics, fluctuations and applications, _Linear Algebra Appl_ 577, 317–359.
  * Mulaik (2009) S. A. Mulaik (2009) _Foundations of factor analysis_ , 2nd edition.  Boca Raton, FL: CRC press.
  * Quadeer _et al._ (2018) A. A. Quadeer, D. Morales-Jimenez & M. R. McKay (2018) Co-evolution networks of HIV/HCV are modular with direct association to structure and function, _PLoS Comput. Biol._ 14 (9), 1–29.
  * SenGupta (1988) A. SenGupta (1988) On loss of power under additional information: An example, _Scand. J. Stat._ 15 (1), 25–31.
  * Soshnikov (2002) A. Soshnikov (2002) A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices, _J. Stat. Phys._ 108 (5), 1033–1056.
  * Van der Vaart (2000) A. W. Van der Vaart (2000) _Asymptotic statistics_.  Cambridge: Cambridge University Press.
  * Yao _et al._ (2015) J. Yao, S. Zheng & Z. Bai (2015) _Large sample covariance matrices and high-dimensional data analysis_.  New York, NY: Cambridge University Press.
  * Yata & Aoshima (2009) K. Yata & M. Aoshima (2009) PCA consistency for non-gaussian data in high dimension, low sample size context, _Commun. Stat. Theory Methods_ 38 (16).
  * Yata & Aoshima (2010) K. Yata & M. Aoshima (2010) Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix, _J. Multivar. Anal._ 101 (9), 2060–2077.
  * Yata & Aoshima (2012) K. Yata & M. Aoshima (2012) Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations, _J. Multivar. Anal._ 105 (1), 193–215.