Central limit theorem for eigenvalue statistics of sample covariance matrix
with random population

Ji Oon Lee

Korea Advanced Institute of Science and Technology
<EMAIL_ADDRESS>
Yiting Li

Korea Advanced Institute of Science and Technology
<EMAIL_ADDRESS>
###### Abstract

Consider the sample covariance matrix

$\displaystyle\Sigma^{1/2}XX^{T}\Sigma^{1/2}$ (1)

where $X$ is an $M\times N$ random matrix with independent entries and
$\Sigma$ is an $M\times M$ diagonal matrix. It is known that if $\Sigma$ is
deterministic, then the fluctuation of

$\sum_{i}f(\lambda_{i})$

converges in distribution to a Gaussian distribution. Here $\\{\lambda_{i}\\}$
are eigenvalues of (1) and $f$ is a good enough test function. In this paper
we consider the case that $\Sigma$ is random and show that the fluctuation of

$\frac{1}{\sqrt{N}}\sum_{i}f(\lambda_{i})$

converges in distribution to a Gaussian distribution. This phenomenon implies
that the randomness of $\Sigma$ decreases the correlation among
$\\{\lambda_{i}\\}$.

###### Contents

  1. 1 Background
  2. 2 Model and main results
  3. 3 Preliminaries
    1. 3.1 A basic tool: $\hat{\mu}_{fc}$
    2. 3.2 Local law
    3. 3.3 Location of extreme eigenvalues
  4. 4 Proof of the central limit theorem

##  1 Background

Consider the sample covariance matrix

$\displaystyle\Sigma^{1/2}XX^{T}\Sigma^{1/2}$ (2)

where $X$ is an $M\times N$ random matrix with iid centered entries and the
population $\Sigma$ is an $M\times M$ diagonal matrix with nonnegative
entries. Assume that the empirical distribution of $\Sigma$ converges to a
deterministic probability measure $\sigma$ and also assume that

$\frac{M}{N}\to\gamma_{0}\in(0,\infty).$

For the case $\Sigma=I$, Marchenko and Pastur [18] proved that the empirical
distribution of (2) converges to a deterministic probability measure with
parameter $\gamma_{0}$. This measure is called the Marchenko-Pastur
distribution and denoted by $\mu_{MP,\gamma_{0}}$. For the case $\Sigma\neq
I$, it is known that empirical distribution of (2) converges to the so called
multiplicative free convolution of $\sigma$ and $\mu_{MP,\gamma_{0}}$, denoted
by $\sigma\boxtimes\mu_{MP,\gamma_{0}}$. See [2, 23, 24].

To know how fast the empirical distribution of (2) converges to
$\sigma\boxtimes\mu_{MP,\gamma_{0}}$, people study the asymptotic behavior of
the fluctuation of the linear statistics:

$\displaystyle\sum_{i}f(\lambda_{i})-M\int
f(t)d(\sigma\boxtimes\mu_{MP,\gamma_{0}})(t)$ (3)

where $\\{\lambda_{i}\\}$ are eigenvalues of (2) and $f$ is a test function.
The central limit theorem (CLT) of (3) was first proved by Johansson [10] for
Wishart matrices where $\Sigma=I$ and $X$ have Gaussian entries. He proved
that (3) converges to a Gaussian distribution. Note that this is different
from the classic CLT for random variables which has the coefficient
$\frac{1}{\sqrt{N}}$. This phenomenon showed that the eigenvalues
$\\{\lambda_{i}\\}$ have very strong correlation. Bai and Silverstein [1]
proved the CLT of (3) for general $X$ and $\Sigma$ with a condition on the
fourth moment of $X_{ij}$. This condition was later removed by the work of Pan
and Zhou [20]. In a series of work [3, 17, 19, 21] the regularity condition of
the test function was weakened. In [16] the CLT was proved in the case when
the entries of $\Sigma$ do not have a uniform bound. In [15] the mesoscopic
CLT was obtained.

All the work cited above assume that $\Sigma$ is a deterministic matrix. Kwak,
the first author and Park [12] considered the largest eigenvalue of (2) for
both deterministic and random $\Sigma$ and proved that the fluctuation of the
largest eigenvalue converges to a deterministic probability measure.

In this paper, we consider the sample covariance matrix with random population
and prove the CLT of (3). In fact, for a good enough test function $f$ we
prove that the rescaled fluctuation

$\displaystyle\frac{1}{\sqrt{N}}\Big{(}\sum_{i}f(\lambda_{i})-M\int
f(t)d(\sigma\boxtimes\mu_{MP,\gamma_{0}})(t)\Big{)}$ (4)

converges in distribution to a centered Gaussian distribution. Different from
the case of deterministic $\Sigma$, we have the coefficient
$\frac{1}{\sqrt{N}}$ in (4). This implies that the randomness of $\Sigma$
decreases the correlation among the eigenvalues.

The similar phenomenon has been observed for the deformed Wigner matrix $W+V$
where $W$ is a Wigner matrix and $V$ is a diagonal matrix. Suppose
$\\{\mu_{i}\\}$ are eigenvalues of $W+V$. Ji and the first author [9] proved
that if $V$ is deterministic, then the fluctuation of

$\displaystyle\sum_{i}f(\mu_{i})$ (5)

converges to a Gaussian distribution and on the other hand, if $V$ is random,
then the fluctuation of (5) multiplied by $\frac{1}{\sqrt{N}}$ converges to a
Gaussian distribution.

##  2 Model and main results

Define

${\mathbb{C}}_{+}=\\{x+{\text{i}}y|x\in{\mathbb{R}},y>0\\}.$

For a probability measure $\pi$, define its Stieltjes transform by

$m_{\pi}(z)=\int\frac{d\pi(t)}{t-z},\quad\forall z\not\in\text{supp}(\pi).$

If $\pi$ has density $\rho_{\pi}(t)$ on an open interval $I$ and
$\rho_{\pi}(t)$ is continuous on $I$, then by the Sokhotski–Plemelj theorem,

$\rho_{\pi}(x)=\frac{1}{\pi}\lim\limits_{y\to
0+}{\text{Im}}m_{\pi}(x+{\text{i}}y)\quad\forall x\in I.$

######  Lemma 1.

Let $\pi$ be a compactly supported probability measure on ${\mathbb{R}}$, and
let $r>0$.

  * •

For each $z\in{\mathbb{C}}_{+}$ there is a unique $m=m(z)\in{\mathbb{C}}_{+}$
satisfying

$\displaystyle\frac{1}{m}=-z+r\int\frac{t}{1+mt}d\pi(t).$ (6)

Moreover, $m(z)$ is the Stieltjes transform of a probability measure with
compact support contained in $[0,\infty)$. This probability measure is the
multiplicative free convolution of $\pi$ and the Marchenko–Pastur law with
dimensional ratio $r$:

$\pi\boxtimes\mu_{MP,r}$

where $\mu_{MP,r}$ denotes the Marchenko–Pastur law with dimensional ratio
$r$.

  * •

There is a continuous nonnegative function $w(x)$ defined on $(0,\infty)$ such
that

$\displaystyle\pi\boxtimes\mu_{MP,r}=(1-r)^{+}\delta_{0}+w(x)dx.$ (7)

Moreover, $w(x)$ is analytic wherever it is positive.

###### Proof.

The first result is Lemma 2.2 in [11]. The second result can be found on Page
2271 of [8]. ∎

######  Definition 1.

Suppose $X=(X_{ij})$ is an $M\times N$ matrix where $M=M(N)$ and

  * •

$\\{X_{ij}\\}$ are iid real-valued random variables such that
${\mathbb{E}}[X_{ij}]=0$, ${\mathbb{E}}[X_{ij}^{2}]=\frac{1}{N}$

  * •

for any $p\in\\{1,2,\ldots\\}$ there exists $C_{p}>0$ such that

$\displaystyle{\mathbb{E}}[|\sqrt{N}X_{ij}|^{p}]\leq C_{p},\quad\forall i,j,N$
(8)

######  Definition 2.

Suppose $l\in(0,1)$ is a constant and $\nu$ is a probability measure with its
density function supported on $[l,1]$. Moreover we assume the density function
is bounded above and below:

$\displaystyle
0<\inf_{x\in[l,1]}\frac{d\nu(x)}{dx}\leq\sup_{x\in[l,1]}\frac{d\nu(x)}{dx}<\infty$
(9)

######  Definition 3.

Let $\Sigma=\text{diag}(\sigma_{1},\ldots,\sigma_{M})$ be an $M\times M$
diagonal matrix such that the entries $\sigma_{1}$,…,$\sigma_{M}$ are iid with
distribution $\nu$. Moreover we assume that $\Sigma$ is independent of $X$.

######  Definition 4.

Suppose there exists $\gamma_{0}\in(0,\infty)\backslash\\{1\\}$ such that

$\displaystyle|\gamma_{0}-\frac{M}{N}|\leq N^{-\frac{1}{2}-c_{0}}$ (10)

where $c_{0}>0$ is a constant. Without loss of generality assume $c_{0}<0.01$.

It is easy to see that if $\gamma_{0}\in(0,1)$, then $M<N$ when $N$ is large
enough; if $\gamma_{0}>1$, then $M>N$ when $N$ is large enough.

By linear algebra, the set of eigenvalues of $X^{T}\Sigma X$ differs from the
set of eigenvalues of $\Sigma^{1/2}XX^{T}\Sigma^{1/2}$ by $|M-N|$ zeros.

Suppose $\mu_{\Sigma}$ is the empirical measure of $\Sigma$:

$\mu_{\Sigma}=\frac{1}{M}\sum_{i=1}^{M}\delta_{\sigma_{i}}.$

######  Lemma 2.

  * •

Almost surely, $\mu_{\Sigma}$ converges in distribution to $\nu$.

  * •

Suppose $\lambda_{1}\geq\cdots\geq\lambda_{N}\geq 0$ are the eigenvalues of
$X^{T}\Sigma X$. Let $\lambda_{i}=0$ for $M\wedge N<i\leq M\vee N$. Almost
surely, the empirical measure

$\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$

converge in distribution to

$\mu_{fc}:=\nu\boxtimes\mu_{MP,\gamma_{0}}.$

###### Proof.

The first conclusion is from the Glivenko–Cantelli theorem. The second
conclusion can be found in the first paragraph of [22]. ∎

######  Definition 5.

  * •

Let $\rho_{fc}(x)$ be the density function of the absolute continuous part of
$\mu_{fc}$. According to Lemma 1, $\rho_{fc}(x)$ is continuous and

$\mu_{fc}=(1-\gamma_{0})^{+}\delta_{0}+\rho_{fc}(x)dx$

  * •

Let $[L_{-},L_{+}]$ be the smallest interval containing the support of
$\rho_{fc}$.

By Proposition 2.4 of [8], $\gamma_{0}\neq 1$ implies $L_{-}>0$.

######  Theorem 1.

Suppose $f(x)$ is a function analytic in a neighborhood of $[L_{-},L_{+}]$.
Then as $N\to\infty$,

$\frac{1}{\sqrt{N}}\sum_{i=1}^{N}f(\lambda_{i})-\sqrt{N}\int f(t)d\mu_{fc}(t)$

converges in distribution to a centered Gaussian distribution whose variance
is

$-\frac{1}{4\pi^{2}}\oint_{\Gamma}\oint_{\Gamma}\int_{l}^{1}\frac{tm_{fc}^{\prime}(\xi_{1})m_{fc}^{\prime}(\xi_{2})f(\xi_{1})f(\xi_{2})}{(1+tm_{fc}(\xi_{1}))(1+tm_{fc}(\xi_{2}))}d\nu(t)d\xi_{1}d\xi_{2}+\Bigg{(}\oint_{\Gamma}f(\xi)m_{fc}^{\prime}(\xi)(\xi+\frac{1}{m_{fc}(\xi)})d\xi\Bigg{)}^{2}$

where $\Gamma$ is a counterclockwise oriented path enclosing $[L_{-},L_{+}]$
but not enclosing 0 such that $f$ is analytic on a neighborhood of the domain
bounded by $\Gamma$.

Since the eigenvalues of $X^{T}\Sigma X$ differ from the eigenvalues of
$\Sigma^{1/2}XX^{T}\Sigma^{1/2}$ by $|M-N|$ zeros, Theorem 1 immediately imply
the central limit theorem of the linear statistics for eigenvalues of
$\Sigma^{1/2}XX^{T}\Sigma^{1/2}$.

##  3 Preliminaries

###  3.1 A basic tool: $\hat{\mu}_{fc}$

######  Definition 6.

Let

$\hat{\mu}_{fc}=\mu_{\Sigma}\boxtimes\mu_{MP,M/N}.$

Let $\hat{\rho}_{fc(t)}$ be the continuous function defined on $(0,\infty)$
such that

$\hat{\mu}_{fc}=(1-\frac{M}{N})^{+}\delta_{0}+\hat{\rho}_{fc}(x)dx.$

(See (7).) Suppose $[\hat{L}_{-},\hat{L}_{+}]$ is the smallest interval
containing the support of $\hat{\rho}_{fc}(x)$.

######  Definition 7.

Let $m_{N}$, $m_{fc}$ and $\hat{m}_{fc}$ denote the Stieltjes transform of
$\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$, $\mu_{fc}$ and
$\hat{\mu}_{fc}$ respectively.

So we have for $z\in{\mathbb{C}}\backslash{\mathbb{R}}$:

$m_{N}(z)=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\lambda_{i}-z}$
$\displaystyle\frac{1}{m_{fc}(z)}=-z+\gamma_{0}\int\frac{t}{1+tm_{fc}(z)}dt,\quad\quad\frac{1}{\hat{m}_{fc}(z)}=-z+\frac{1}{N}\sum_{i=1}^{M}\frac{\sigma_{i}}{1+\sigma_{i}\hat{m}_{fc}(z)}$
(11)

######  Lemma 3.

For any constant $\epsilon>0$, we have

${\mathbb{P}}(|\hat{L}_{-}-L_{-}|<\epsilon\quad\text{and}\quad|\hat{L}_{+}-L_{+}|<\epsilon)\to
1\quad\text{as }N\to\infty$

###### Proof.

Suppose $x_{0}$ is the unique number in $(0,1)$ such that
$h_{0}(x_{0})=\frac{1}{\gamma_{0}}$ where

$h_{0}(x):=\int_{l}^{1}\Big{(}\frac{xt}{1-xt}\Big{)}^{2}d\nu(t).$

According to Proposition 2.3 of [8], we have

$L_{+}=\frac{1}{x_{0}}+\gamma_{0}\int_{l}^{1}\frac{t}{1-x_{0}t}d\nu(t)$

On the other hand, suppose $x_{1}$ is the unique number in
$(0,\frac{1}{\max_{i}\sigma_{i}})$ such that $h_{1}(x_{1})=\frac{N}{M}$ where

$h_{1}(x):=\frac{1}{M}\sum_{i=1}^{M}\Big{(}\frac{x\sigma_{i}}{1-x\sigma_{i}}\Big{)}^{2}$

By Lemma 2.4–2.6 of [11],

$\hat{L}_{+}=\frac{1}{x_{1}}+\frac{1}{N}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-x_{1}\sigma_{i}}$

Note that $x_{0}$ is deterministic and is independent of $N$, but $x_{1}$ is
random and depends on $N$.

Suppose $c>0$ is a small enough constant such that
$[x_{0}-c,x_{0}+c]\subset(0,1)$. We claim that

$\displaystyle{\mathbb{P}}(x_{1}\in[x_{0}-c,x_{0}+c])\to 1.$ (12)

In fact, since $h_{1}$ is increasing on $[0,1]$, there exists $a>0$ such that

$h_{0}(x_{0}-c)+a<h_{0}(x_{0})<h_{0}(x_{0}+c)-a.$

Note that both $\sqrt{M}(h_{0}(x_{0}-c)-h_{1}(x_{0}-c))$ and
$\sqrt{M}(h_{0}(x_{0}+c)-h_{1}(x_{0}+c))$ converge in distribution to a
Gaussian distribution, so

$\displaystyle{\mathbb{P}}\Big{(}|h_{0}(x_{0}\pm c)-h_{1}(x_{0}\pm
c)|<a\Big{)}\to 1.$ (13)

When $|h_{0}(x_{0}\pm c)-h_{1}(x_{0}\pm c)|<a$ holds, we have

$h_{1}(x_{0}-c)<h_{0}(x_{0}-c)+a<h_{0}(x_{0})<h_{0}(x_{0}+c)-a<h_{1}(x_{0}+c)$

Note that $|\frac{1}{\gamma_{0}}-\frac{N}{M}|\to 0$. So if $N$ is large enough
and $|h_{0}(x_{0}\pm c)-h_{1}(x_{0}\pm c)|<a$ holds,

$h_{1}(x_{0}-c)<h_{1}(x_{1})<h_{1}(x_{0}+c)$

thus by monotonicity of $h_{1}$:

$\displaystyle x_{0}-c\leq x_{1}\leq x_{0}+c\quad\text{whenever
}|h_{0}(x_{0}\pm c)-h_{1}(x_{0}\pm c)|<a$ (14)

(13) and (14) prove (12).

Now we prove the lemma. First, choose $\delta$ small enough such that
$[x_{0}-\delta,x_{0}+\delta]\in(0,1)$ and

$\displaystyle\Big{|}\gamma_{0}\int_{l}^{1}\frac{t}{1-x_{0}t}d\nu(t)-\gamma_{0}\int_{l}^{1}\frac{t}{1-(x_{0}\pm\delta)t}d\nu(t)\Big{|}<\epsilon$
(15)

and

$|\frac{1}{x_{0}}-\frac{1}{x_{0}\pm\delta}|<\epsilon.$

Since
$\frac{1}{\sqrt{M}}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-(x_{0}\pm\delta)\sigma_{i}}-\sqrt{M}\int_{l}^{1}\frac{t}{1-(x_{0}\pm\delta)t}d\nu(t)$
converges in distribution to a normal distribution, we have

$\displaystyle{\mathbb{P}}\Big{(}\Big{|}\frac{1}{M}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-(x_{0}\pm\delta)\sigma_{i}}-\int_{l}^{1}\frac{t}{1-(x_{0}\pm\delta)t}d\nu(t)\Big{|}<\epsilon\Big{)}\to
1.$ (16)

By (12) and (16), the event

$\displaystyle\\{|x_{0}-x_{1}|<\delta\\}\cap\Big{\\{}\Big{|}\frac{1}{M}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-(x_{0}\pm\delta)\sigma_{i}}-\int_{l}^{1}\frac{t}{1-(x_{0}\pm\delta)t}d\nu(t)\Big{|}<\epsilon\Big{\\}}$
(17)

has a probability going to 1. On (17), by (15) we have

$\int\frac{t}{1-x_{0}t}d\nu(t)-2\epsilon<\frac{1}{M}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-x_{1}\sigma_{i}}<\int\frac{t}{1-x_{0}t}d\nu(t)+2\epsilon$

and then for large enough $N$

$|L_{+}-\hat{L}_{+}|\leq|\frac{1}{x_{1}}-\frac{1}{x_{0}}|+\Big{|}\frac{1}{N}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-x_{1}\sigma_{i}}-\gamma_{0}\int\frac{t}{1-x_{0}t}d\nu(t)\Big{|}\\\
\leq|\frac{1}{x_{0}+\delta}-\frac{1}{x_{0}}|\vee|\frac{1}{x_{0}-\delta}-\frac{1}{x_{0}}|+\frac{M}{N}\Big{|}\frac{1}{M}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-x_{1}\sigma_{i}}-\int\frac{t}{1-x_{0}t}d\nu(t)\Big{|}+|\frac{M}{N}-\gamma_{0}|\int\frac{t}{1-x_{0}t}d\nu(t)\\\
\leq\epsilon+\frac{M}{N}\cdot 2\epsilon+\epsilon\quad\text{(since
$|\frac{M}{N}-\gamma_{0}|$ goes to 0)}$

Recall ${\mathbb{P}}(\eqref{eqn13})\to 1$, so we have proved

${\mathbb{P}}(|L_{+}-\hat{L}_{+}|\leq 2\epsilon(1+\frac{M}{N}))\to 1.$

Similarly, by Lemma 2.4–2.6 of [11] and Proposition 2.3 of [8] we have the
following.

  * •

$\hat{L}_{-}=\frac{1}{x_{1}^{\prime}}+\frac{1}{N}\sum_{i=1}^{M}\frac{\sigma_{i}}{1-x_{1}^{\prime}\sigma_{i}}$
where $x_{1}^{\prime}$ is the unique number in
$(-\infty,0)\cup(\frac{1}{\min_{i}\sigma_{i}},\infty)$ such that
$h_{1}(x_{1}^{\prime})=\frac{N}{M}$. In particular,
$x_{1}^{\prime}\in(-\infty,0)$ when $\gamma_{0}>1$,
$x_{1}^{\prime}\in(\frac{1}{\min_{i}\sigma_{i}},\infty)$ when
$\gamma_{0}\in(0,1)$.

  * •

$L_{-}=\frac{1}{x_{0}^{\prime}}+\gamma_{0}\int_{l}^{1}\frac{t}{1-x_{0}t}d\nu(t)$
where $x_{0}^{\prime}$ is the unique number in
$(-\infty,0)\cup(\frac{1}{l},\infty)$ such that
$h_{0}(x_{0}^{\prime})=\frac{1}{\gamma_{0}}$. In particular,
$x_{0}^{\prime}\in(-\infty,0)$ when $\gamma_{0}>1$,
$x_{0}^{\prime}\in(\frac{1}{l},\infty)$ when $\gamma_{0}\in(0,1)$.

Then using same argument as above we can prove

${\mathbb{P}}(|L_{-}-\hat{L}_{-}|\leq 2\epsilon(1+\frac{M}{N}))\to 1.$

Since $\frac{M}{N}$ is bounded and $\epsilon$ can be arbitrarily small, the
lemma is proved.

∎

###  3.2 Local law

######  Definition 8.

  * •

Suppose $\Sigma^{a}=\text{Diag}(a_{1},\ldots,a_{M})$ is an $M\times M$
deterministic diagonal matrix such that $a_{1},\ldots,a_{M}$ are all in
$[l,1]$.

  * •

Set

$\mu_{\Sigma^{a}}=\frac{1}{M}\sum_{i=1}^{M}\delta_{a_{i}}\quad\text{and}\quad\mu_{fc}^{a}=\mu_{\Sigma^{a}}\boxtimes\mu_{MP,M/N}.$

Suppose $[L_{-}^{a},L_{+}^{a}]$ is the smallest interval containing the
support of the absolute continuous part of $\mu_{fc}^{a}$. (See (7).)

  * •

Suppose $\lambda_{1}^{a}\geq\cdots\geq\lambda_{N}^{a}\geq 0$ are eigenvalues
of $X^{T}\Sigma^{a}X$. Let

$m_{N}^{a}(z)=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\lambda_{i}^{a}-z},\quad\quad
m_{fc}^{a}(z)=\int\frac{1}{t-z}d\mu_{fc}^{a}(t)$

  * •

Suppose $\tau\in(0,1)$ is a constant. Let

$D_{\tau}=\Big{\\{}z=E+{\text{i}}\eta\in{\mathbb{C}}_{+}\Big{|}|z|\geq\tau,|E|\leq\tau^{-1},N^{-1+\tau}\leq\eta\leq\tau^{-1}\Big{\\}}.$

The following theorem is obtained by Knowles and Yin [11].

######  Theorem 2.

Suppose $\mu_{\Sigma^{a}}$ converges in distribution to $\nu$:

$\displaystyle\mu_{\Sigma^{a}}\Longrightarrow\nu.$ (18)

For any $\epsilon>0$, $D>0$, if $N$ is large enough then

${\mathbb{P}}\Big{(}|m_{N}^{a}(z)-m_{fc}^{a}(z)|\leq\frac{N^{\epsilon}}{N{\text{Im}}z}\Big{)}>1-N^{-D},\quad\forall
z\in D_{\tau}$ $\displaystyle{\mathbb{P}}\Big{(}\lambda_{1}^{a}\leq
L_{+}^{a}+N^{-\frac{2}{3}+\epsilon}\quad\text{and}\quad\lambda_{M\wedge
N}>L_{-}^{a}-N^{-\frac{2}{3}+\epsilon}\Big{)}>1-N^{-D}$ (19)

###### Proof.

The first conclusion comes from Theorem 3.6 of [11]. The second conclusion is
a direct corollary of Theorem 3.12 of [11]. We remark that (18) ensures that
the regularity conditions in the theorems in [11] are satisfied. See
Definition 2.7 and Example 2.9 of [11]. ∎

The above theorem immediately induce the following local law for our main
model.

######  Theorem 3.

For any $\epsilon>0$ and $D>0$, if $N>N_{0}(\epsilon,D,\tau)$ then

${\mathbb{P}}\Big{(}|m_{N}(z)-\hat{m}_{fc}(z)|\leq\frac{N^{\epsilon}}{N{\text{Im}}z}\quad\forall
z\in D_{\tau}\Big{)}>1-N^{-D}$

###### Proof.

Suppose $z\in D_{\tau}$. Set

$w(X,\Sigma,z)=|m_{N}(z)-\hat{m}_{fc}(z)|\frac{N{\text{Im}}z}{N^{\epsilon}}.$

Let $\sigma(\Sigma)$ be the sigma algebra generated by entries of $\Sigma$ and
let $\omega$ denote a sample point in the probability space. Then

${\mathbb{P}}\Big{(}|m_{N}(z)-\hat{m}_{fc}(z)|\leq\frac{N^{\epsilon}}{N{\text{Im}}z}\Big{)}={\mathbb{P}}\Big{(}w(X,\Sigma,z)\leq
1\Big{)}={\mathbb{E}}[{\mathbb{E}}[\mathds{1}_{w(X,\Sigma,z)\leq
1}|\sigma(\Sigma)]]\\\ =\int{\mathbb{E}}[\mathds{1}_{w(X,\Sigma,z)\leq
1}|\sigma(\Sigma)](\omega)d{\mathbb{P}}(\omega)=\int{\mathbb{E}}[\mathds{1}_{w(X,\Sigma(\omega),z)\leq
1}]d{\mathbb{P}}(\omega)=\int{\mathbb{P}}(w(X,\Sigma(\omega),z)\leq
1)d{\mathbb{P}}(\omega)$

where we used the independence of $X$ and $\Sigma$ in the fourth identity.

By Lemma 2, almost surely, $\Sigma$ satisfies the conditions of $\Sigma^{a}$
in Theorem 2, so we have for $N>N_{0}(\epsilon,D,\tau)$,
${\mathbb{P}}(w(X,\Sigma(\omega),z)\leq 1)>1-N^{-D}$ and

${\mathbb{P}}\Big{(}|m_{N}(z)-\hat{m}_{fc}(z)|\leq\frac{N^{\epsilon}}{N{\text{Im}}z}\Big{)}=\int{\mathbb{P}}(w(X,\Sigma(\omega),z)\leq
1)d{\mathbb{P}}(\omega)>1-N^{-D}$

Then by a classical “lattice” argument we complete the proof. ∎

###  3.3 Location of extreme eigenvalues

######  Lemma 4.

For any $\epsilon>0$ we have

${\mathbb{P}}\Big{(}\lambda_{1},\ldots,\lambda_{M\wedge N}\text{ are all in
}[L_{-}-\epsilon,L_{+}+\epsilon]\Big{)}\to 1.$

###### Proof.

Similarly as in Theorem 3, let $\sigma(\Sigma)$ be the sigma algebra generated
by entries of $\Sigma$ and

$w(X,\Sigma)=\mathds{1}_{[\lambda_{M\wedge
N},\lambda_{1}]\subset[\hat{L}_{-}-0.5\epsilon,\hat{L}_{+}+0.5\epsilon]}.$

Using (19), we have for $N>N_{0}(\epsilon,D,\tau)$,

${\mathbb{P}}\Big{(}[\lambda_{M\wedge
N},\lambda_{1}]\subset[\hat{L}_{-}-0.5\epsilon,\hat{L}_{+}+0.5\epsilon]\Big{)}={\mathbb{E}}[w(X,\Sigma)]={\mathbb{E}}[{\mathbb{E}}[w(X,\Sigma)|\sigma(\Sigma)]]\\\
=\int{\mathbb{E}}[w(X,\Sigma(\omega))]d{\mathbb{P}}(\omega)>1-N^{-D}$

This together with Lemma 3 complete the proof.

∎

##  4 Proof of the central limit theorem

In this section we prove Theorem 1. First of all we suppose

  * •

$d\in(0,\frac{L_{-}}{10})$ is a small constant such that $f$ is analytic on a
neighborhood of the closed rectangle with vertices $L_{+}+2d\pm 2d{\text{i}}$
and $L_{-}-2d\pm 2d{\text{i}}$;

  * •

$\Gamma$ is the boundary of the rectangle described above with
counterclockwise orientation;

  * •

$\Omega_{N}=\big{\\{}|\hat{L}_{-}-L_{-}|<d\quad\text{and}\quad|\hat{L}_{+}-L_{+}|<d\big{\\}}$
$\tilde{\Omega}_{N}=\big{\\{}\lambda_{1},\ldots,\lambda_{M\wedge N}\text{ are
all in }[L_{-}-d,L_{+}+d]\big{\\}}\cap\Omega_{N}$

By Lemma 3 and Lemma 4, we have

${\mathbb{P}}(\tilde{\Omega}_{N})\to 1.$

Suppose $\tilde{\Omega}_{N}$ holds, then we have
$\lambda_{i}=\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{1}{\xi-\lambda_{i}}d\xi$
for $1\leq i\leq M\wedge N$, and:

$\frac{1}{\sqrt{N}}\sum_{i=1}^{N}f(\lambda_{i})-\sqrt{N}\int
f(t)d\mu_{fc}(t)\\\
=\frac{1}{\sqrt{N}}\Bigg{[}\frac{1}{2\pi{\text{i}}}\sum_{i=1}^{M\wedge
N}\oint_{\Gamma}\frac{f(\xi)d\xi}{\xi-\lambda_{i}}-\frac{N}{2\pi{\text{i}}}\int\rho_{fc}(t)\oint_{\Gamma}\frac{f(\xi)d\xi}{\xi-t}dt+f(0)[(N-M)^{+}-N(1-\gamma_{0})^{+}]\Bigg{]}\\\
=\frac{1}{\sqrt{N}2\pi{\text{i}}}\oint_{\Gamma}f(\xi)\Bigg{[}N\int\frac{\rho_{fc}(t)}{t-\xi}dt-\sum_{i=1}^{M\wedge
N}\frac{1}{\lambda_{i}-\xi}\Bigg{]}d\xi+\frac{f(0)}{\sqrt{N}}[(N-M)^{+}-N(1-\gamma_{0})^{+}]\\\
=\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(m_{fc}(\xi)-m_{N}(\xi))d\xi+\frac{f(0)}{\sqrt{N}}[(N-M)^{+}-N(1-\gamma_{0})^{+}]$
(20)

where we used

$\oint_{\Gamma}f(\xi)\int\frac{\rho_{fc}(t)}{t-\xi}dtd\xi=\oint_{\Gamma}f(\xi)\Big{(}\int\frac{d\mu_{fc}(t)}{t-\xi}+\frac{(1-\gamma_{0})^{+}}{\xi}\Big{)}d\xi=\oint_{\Gamma}f(\xi)\int\frac{d\mu_{fc}(t)}{t-\xi}d\xi$

and

$\oint_{\Gamma}f(\xi)\sum_{i=1}^{M\wedge
N}\frac{1}{\lambda_{i}-\xi}d\xi=\oint_{\Gamma}f(\xi)\Big{(}\sum_{i=1}^{N}\frac{1}{\lambda_{i}-\xi}+\frac{(N-M)^{+}}{\xi}\Big{)}d\xi=\oint_{\Gamma}f(\xi)\sum_{i=1}^{N}\frac{1}{\lambda_{i}-\xi}d\xi$

in the last step of (20).

Note that ${\mathbb{P}}(\tilde{\Omega}_{N})\to 1$ and

$\Big{|}\frac{f(0)}{\sqrt{N}}[(N-M)^{+}-N(1-\gamma_{0})^{+}]\Big{|}\leq
f(0)\sqrt{N}|\gamma_{0}-\frac{M}{N}|\to 0\quad\text{(by \eqref{eqn18})}.$

So it suffices to prove that

$\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(m_{fc}(\xi)-m_{N}(\xi))d\xi=\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(\hat{m}_{fc}(\xi)-m_{N}(\xi))d\xi+\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(m_{fc}(\xi)-\hat{m}_{fc}(\xi))d\xi$

converges in distribution to the limiting distribution in the statement of
Theorem 1.

When $\tilde{\Omega}_{N}$ holds, we have by definition:

$\text{dist}(\xi,\text{supp}(\hat{m}_{fc}))\geq
d\quad\text{and}\quad\text{dist}(\xi,\lambda_{i})\geq
d\quad\forall\xi\in\Gamma,i\in\\{1,\ldots,N\\}$

and then

$|m_{N}(\xi)|\leq d^{-1},\quad|\hat{m}_{fc}(\xi)|\leq
d^{-1},\quad\forall\xi\in\Gamma.$

Choose $\tau\in(0,0.1)$ small enough such that $\Gamma\subset D_{\tau}$.

By Theorem 3 with $\epsilon=0.1$ and $D=1$, if $N>N_{0}(\tau,d)$, then with
probability larger than $1-N^{-1}$ we have

$\Big{|}\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(\hat{m}_{fc}(\xi)-m_{N}(\xi))f(\xi)d\xi\Big{|}\\\
\leq\Big{|}\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\int_{\Gamma\cap\\{|{\text{Im}}\xi|<N^{-1+\tau}\\}}(\hat{m}_{fc}(\xi)-m_{N}(\xi))f(\xi)d\xi\Big{|}+\Big{|}\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\int_{\Gamma\cap\\{|{\text{Im}}\xi|>N^{-1+\tau}\\}}(\hat{m}_{fc}(\xi)-m_{N}(\xi))f(\xi)d\xi\Big{|}\\\
\leq\Big{(}\frac{\sqrt{N}}{2\pi}4N^{-1+\tau}\cdot\frac{2}{d}+\frac{\sqrt{N}}{2\pi}\int_{\Gamma\cap\\{|{\text{Im}}\xi|>N^{-1+\tau}\\}}\frac{N^{-0.9}}{|{\text{Im}}\xi|}d\xi\Big{)}\sup_{\xi\in\Gamma}|f(\xi)|\\\
\leq\big{(}\frac{2}{d}N^{-0.5+\tau}+N^{-0.4}\ln
N\big{)}\sup_{\xi\in\Gamma}|f(\xi)|$

So
$\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(\hat{m}_{fc}(\xi)-m_{N}(\xi))f(\xi)d\xi$
converges in distribution to 0 and now we only need to show that

$\mathds{1}_{\tilde{\Omega}_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(m_{fc}(\xi)-\hat{m}_{fc}(\xi))f(\xi)d\xi$

or

$\mathds{1}_{\Omega_{N}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(m_{fc}(\xi)-\hat{m}_{fc}(\xi))f(\xi)d\xi$

converges in distribution to the limiting distribution in the statement of
Theorem 1.

######  Lemma 5.

There exists a constant $C_{d}>0$ depending on $d$ such that

  * •

$|1+tm_{fc}(z)|>C_{d}\quad\forall z\in\Gamma,t\in[l,1]$

  * •

if $\Omega_{N}$ holds then

$|1+\sigma_{i}\hat{m}_{fc}(z)|>C_{d}\quad\forall z\in\Gamma,1\leq i\leq M$

###### Proof.

Suppose $\Omega_{N}$ holds. According to Lemma 2.5 of [11],

$-x_{1}=\hat{m}_{fc}(\hat{L}_{+}):=\lim_{\eta\to
0+}\hat{m}_{fc}(\hat{L}_{+}+{\text{i}}\eta)$

where $x_{1}\in(0,\frac{1}{\max_{i}\sigma_{i}})$ is defined in the proof of
Lemma 3. It’s easy to see $\hat{m}_{fc}$ is negative and is increasing on
$[\hat{L}_{+},\infty)$, so

$|\hat{m}_{fc}(L_{+}+2d)-\hat{m}_{fc}(\hat{L}_{+})|>|\hat{m}_{fc}(L_{+}+2d)-\hat{m}_{fc}(L_{+}+d)|>\frac{d}{(L_{+}+2d)^{2}}$

which implies

$|\hat{m}_{fc}(L_{+}+2d)|<x_{1}-\frac{d}{(L_{+}+2d)^{2}}<\frac{1}{\max_{i}\sigma_{i}}-\frac{d}{(L_{+}+2d)^{2}}$

and thus

$\displaystyle|1+\sigma_{i}\hat{m}_{fc}(L_{+}+2d)|>\frac{ld}{(L_{+}+2d)^{2}}$
(21)

If $\gamma_{0}>1$ then $\hat{\mu}_{fc}$ does not have an atom at 0, so
$\hat{m}_{fc}>0$ on $(-\infty,\hat{L}_{-})$ and

$\displaystyle|1+\sigma_{i}\hat{m}_{fc}(L_{-}-2d)|>1$ (22)

Suppose $\gamma\in(0,1)$. According to Lemma 2.5 of [11],

$-x_{1}^{\prime}=\hat{m}_{fc}(\hat{L}_{-}):=\lim_{\eta\to
0+}\hat{m}_{fc}(\hat{L}_{-}+{\text{i}}\eta)$

where $x_{1}^{\prime}\in(\frac{1}{\min_{i}\sigma_{i}},\infty)$ is defined in
the proof of Lemma 3. It’s easy to see $\hat{m}_{fc}$ is is increasing on
$(0,\hat{L}_{-}]$, so

$|\hat{m}_{fc}(L_{-}-2d)-\hat{m}_{fc}(\hat{L}_{-})|>|\hat{m}_{fc}(L_{-}-2d)-\hat{m}_{fc}(L_{-}-d)|>\frac{d}{(L_{+}+2d)^{2}}$

which implies

$\hat{m}_{fc}(L_{-}-2d)<-x_{1}^{\prime}-\frac{d}{(L_{+}+2d)^{2}}<-\frac{1}{\min_{i}\sigma_{i}}-\frac{d}{(L_{+}+2d)^{2}}$

and thus

$\displaystyle|1+\sigma_{i}\hat{m}_{fc}(L_{-}-2d)|>\frac{ld}{(L_{+}+2d)^{2}}.$
(23)

By (21), (22), (23) and the continuity of $\hat{m}_{fc}$, there exist
constants $c_{1}>0$ and $c_{2}>0$ both determined by $d$ such that if
$|\eta|<c_{1}$ then

$|1+\sigma_{i}\hat{m}_{fc}(L_{+}+2d+{\text{i}}\eta)|>c_{2}$
$|1+\sigma_{i}\hat{m}_{fc}(L_{-}-2d+{\text{i}}\eta)|>c_{2}.$

On the other hand, if $z\in\Gamma$ with $|{\text{Im}}z|\geq c_{1}$ then

$|1+\sigma_{i}\hat{m}_{fc}(z)|>\sigma_{i}|{\text{Im}}\hat{m}_{fc}(z)|\geq
lc_{1}\int\frac{d\hat{\mu}_{fc}(t)}{|t-z|^{2}}>c_{3}$

where $c_{3}$ is determined by $d$.

So we proved the first conclusion of the lemma. The second conclusion can be
proved in the same way, with the properties introduced in the proof of Lemma
3.

∎

######  Definition 9.

For $z\in{\mathbb{C}}\backslash{\mathbb{R}}$ and $1\leq i\leq M$ we define

$g_{i}(z)=\frac{\sigma_{i}}{1+\sigma_{i}m_{fc}(z)},\quad\hat{g}_{i}(z)=\frac{\sigma_{i}}{1+\sigma_{i}\hat{m}_{fc}(z)}$

######  Lemma 6.

Suppose $a_{1}>0$, $a_{2}>0$ are constants. Then

${\mathbb{P}}\Big{(}\Big{|}\frac{1}{N^{\frac{1}{2}+a_{1}}}\sum_{i=1}^{N}(g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)])\Big{|}\leq
a_{2},\quad\forall\xi\in\Gamma\Big{)}\to 1\quad\text{as }N\to\infty$
${\mathbb{P}}\Big{(}\Big{|}\frac{1}{N^{\frac{1}{2}+a_{1}}}\sum_{i=1}^{N}(g_{i}^{2}(\xi)-{\mathbb{E}}[g_{i}^{2}(\xi)])\Big{|}\leq
a_{2},\quad\forall\xi\in\Gamma\Big{)}\to 1\quad\text{as }N\to\infty$

###### Proof.

It can be proved in the same way as Lemma 19 in [13]. Note that $g_{i}$ are
bounded functions thanks to Lemma 5. ∎

######  Lemma 7.

Suppose $z\in{\mathbb{C}}\backslash{\mathbb{R}}$. Then we have

$\displaystyle\hat{m}_{fc}-m_{fc}=A_{N}(\hat{m}_{fc}-m_{fc})^{2}+B_{N}$ (24)

where

$\displaystyle
A_{N}=\frac{m_{fc}^{\prime}}{m_{fc}}\frac{M}{N}\int\Big{(}\frac{t}{1+tm_{fc}}\Big{)}^{2}d\nu(t)-\frac{m_{fc}^{\prime}\hat{m}_{fc}}{Nm_{fc}}\sum(g_{i}^{2}\hat{g}_{i})$
$B_{N}=-\frac{m_{fc}^{\prime}}{N}\sum(g_{i}-{\mathbb{E}}g_{i})+\frac{m_{fc}^{\prime}\hat{m}_{fc}}{Nm_{fc}}(\hat{m}_{fc}-m_{fc})\sum(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])-\frac{m_{fc}^{\prime}\hat{m}_{fc}}{m_{fc}}(\frac{M}{N}-\gamma_{0})\int\frac{td\nu(t)}{1+tm_{fc}}\\\
+m_{fc}^{\prime}(\hat{m}_{fc}-m_{fc})\int\frac{t^{2}d\nu(t)}{(1+tm_{fc})^{2}}(\frac{M}{N}-\gamma_{0})+\frac{m_{fc}^{\prime}(m_{fc}-\hat{m}_{fc})}{Nm_{fc}}\sum(g_{i}-{\mathbb{E}}g_{i})$

where each sum is over $i=1,\ldots,M$.

###### Proof.

By (11),

$\displaystyle\hat{m}_{fc}-m_{fc}=\hat{m}_{fc}m_{fc}(\frac{1}{m_{fc}}-\frac{1}{\hat{m}_{fc}})=-\hat{m}_{fc}m_{fc}\Big{[}\frac{1}{N}\sum_{i=1}^{M}(\hat{g}_{i}-{\mathbb{E}}g_{i})+(\frac{M}{N}-\gamma_{0})\int\frac{td\nu(t)}{1+tm_{fc}}\Big{]}$
(25)

Note that

$\sum_{i=1}^{M}(\hat{g}_{i}-{\mathbb{E}}g_{i})=\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+\sum_{i=1}^{M}(\hat{g}_{i}-g_{i})=\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+(m_{fc}-\hat{m}_{fc})\sum_{i=1}^{M}g_{i}\hat{g}_{i}\\\
=\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+(m_{fc}-\hat{m}_{fc})(\sum_{i=1}^{M}g_{i}^{2}+(m_{fc}-\hat{m}_{fc})\sum_{i=1}^{M}g_{i}^{2}\hat{g}_{i})\\\
=\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+(m_{fc}-\hat{m}_{fc})\sum_{i=1}^{M}g_{i}^{2}+(m_{fc}-\hat{m}_{fc})^{2}\sum_{i=1}^{M}g_{i}^{2}\hat{g}_{i}\\\
=\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+(m_{fc}-\hat{m}_{fc})\sum_{i=1}^{M}(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])+(m_{fc}-\hat{m}_{fc})^{2}\sum_{i=1}^{M}g_{i}^{2}\hat{g}_{i}-M(\hat{m}_{fc}-m_{fc}){\mathbb{E}}[g_{i}^{2}].$
(26)

Plugging (26) into (25), we have

$\hat{m}_{fc}-m_{fc}=-\frac{m_{fc}\hat{m}_{fc}}{N}\sum_{i=1}^{M}(g_{i}-{\mathbb{E}}g_{i})+\frac{m_{fc}\hat{m}_{fc}}{N}(\hat{m}_{fc}-m_{fc})\sum_{i=1}^{M}(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])\\\
-\frac{m_{fc}\hat{m}_{fc}}{N}(\hat{m}_{fc}-m_{fc})^{2}\sum_{i=1}^{M}g_{i}^{2}\hat{g}_{i}-\hat{m}_{fc}m_{fc}(\frac{M}{N}-\gamma_{0})\int\frac{td\nu(t)}{1+tm_{fc}}+\frac{M}{N}\hat{m}_{fc}m_{fc}(\hat{m}_{fc}-m_{fc}){\mathbb{E}}[g_{i}^{2}]$
(27)

In (27), we replace the last term by

$m_{fc}(\hat{m}_{fc}-m_{fc}){\mathbb{E}}[g_{i}^{2}]\Big{(}\frac{M}{N}(\hat{m}_{fc}-m_{fc})+m_{fc}(\frac{M}{N}-\gamma_{0})\Big{)}+(\hat{m}_{fc}-m_{fc})m_{fc}^{2}\gamma_{0}{\mathbb{E}}[g_{i}^{2}],$

then move the term
$(\hat{m}_{fc}-m_{fc})m_{fc}^{2}\gamma_{0}{\mathbb{E}}[g_{i}^{2}]$ to the left
hand side, then use the fact that

$\displaystyle
1-(\hat{m}_{fc}-m_{fc})m_{fc}^{2}\gamma_{0}{\mathbb{E}}[g_{i}^{2}]=1-\gamma_{0}m_{fc}\int\frac{t^{2}d\nu(t)}{(1+tm_{fc})^{2}}=\frac{m_{fc}^{2}}{m_{fc}^{\prime}}$
(28)

then we obtain (24). Here we used the definition of $g_{i}$ in the first
identity of (28) and used the self-consistent equation (6) to find the
$m_{fc}^{\prime}$ for the second identity of (28).

∎

Let

$\Omega_{N}^{\prime}=\Omega_{N}\cap\Big{\\{}\Big{|}\frac{1}{N^{\frac{1}{2}+0.1c_{0}}}\sum_{i=1}^{N}(g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)])\Big{|}\leq
1,\Big{|}\frac{1}{N^{\frac{1}{2}+0.1c_{0}}}\sum_{i=1}^{N}(g_{i}^{2}(\xi)-{\mathbb{E}}[g_{i}^{2}(\xi)])\Big{|}\leq
1\quad\forall\xi\in\Gamma\Big{\\}}$

where $c_{0}\in(0,0.01)$ is defined in Definition 4. Then
${\mathbb{P}}(\Omega^{\prime})\to 1$. On $\Omega^{\prime}$ we have for each
$z\in\Gamma$:

$|A_{N}|\leq C_{1},\quad|B_{N}|\leq C_{1}N^{-0.5+0.1c_{0}}$

where $C_{1}>0$ is a constant determined by $d$. By Lemma 7, on
$\Omega_{N}^{\prime}$ we have for $z\in\Gamma$

$\hat{m}_{fc}-m_{fc}=\frac{1+x_{1}}{2A_{N}}\quad\text{or}\quad\hat{m}_{fc}-m_{fc}=\frac{1+x_{2}}{2A_{N}}$

where $x_{1,2}$ are square roots of $1-4A_{N}B_{N}$ such that $x_{1}$ is close
to $-1$ and $x_{2}$ is close to $1$. So for $N>N_{0}(d)$ and $z\in\Gamma$

$|1+x_{1}|\leq 5C_{1}^{2}N^{-0.5+0.1c_{0}},\quad|1-x_{2}|\leq
5C_{1}^{2}N^{-0.5+0.1c_{0}}.$

If $\Omega_{N}^{\prime}$ holds and $N>N_{0}(d)$ then

$\Big{|}\frac{1+x_{1}}{2A_{N}}-\frac{1+x_{2}}{2A_{N}}\Big{|}\geq\frac{1}{2C_{1}},\forall
z\in\Gamma$

so by continuity, we have

$\displaystyle\hat{m}_{fc}-m_{fc}=\frac{1+x_{1}}{2A_{N}},\forall z\in\Gamma$
(29)

or

$\displaystyle\hat{m}_{fc}-m_{fc}=\frac{1+x_{2}}{2A_{N}},\forall z\in\Gamma$
(30)

Suppose $z_{0}\in\Gamma$ and ${\text{Im}}z_{0}=1$, by Lemma 2 and Theorem 3,
we see
$m_{fc}(z_{0})-\hat{m}_{fc}(z_{0})=m_{fc}(z_{0})-m_{N}(z_{0})+m_{N}(z_{0})-\hat{m}_{fc}(z_{0})$
converges in probability to 0. Note that if $N>N_{0}(d)$, then
$\quad|\frac{1+x_{2}}{2A_{N}(z_{0})}|\geq\frac{1}{2C_{1}}$ holds on
$\Omega_{N}^{\prime}$, so (30) cannot happen, so we have (29). Plugging (29)
into (24), we see that if $N>N_{0}(d)$, $\Omega_{N}^{\prime}$ holds and
$z\in\Gamma$ then

$(1+x_{1})(1-x_{1})=4A_{N}B_{N}$

and

$\displaystyle|\hat{m}_{fc}-m_{fc}|=|\frac{1+x_{1}}{2A_{N}}|\leq|\frac{2B_{N}}{1-x_{1}}|\leq|2B_{N}|\leq
2C_{1}N^{-0.5+0.1c_{0}}$ (31)

Putting (31) into the definitions of $B_{N}$ we see that there exists a
constant $C_{2}=C_{2}(d)>0$ such that if $\Omega_{N}^{\prime}$ holds and
$N>N_{0}(d)$ then

$\displaystyle
B_{N}=-\frac{m_{fc}^{\prime}}{N}\sum(g_{i}-{\mathbb{E}}g_{i})+R_{N}$

where

$|R_{N}|\leq C_{2}N^{-\frac{1}{2}-c_{0}}$

and therefore by (24), (31) and estimation of $A_{N}$,

$\hat{m}_{fc}-m_{fc}=-\frac{m_{fc}^{\prime}}{N}\sum(g_{i}-{\mathbb{E}}g_{i})+W_{N}$

where

$|W_{N}|\leq 4C_{1}^{3}N^{-1+0.2c_{0}}+C_{2}N^{-\frac{1}{2}-c_{0}}.$

So, for $N>N_{0}(d)$,

$\Big{|}\mathds{1}_{\Omega_{N}^{\prime}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(m_{fc}(\xi)-\hat{m}_{fc}(\xi))f(\xi)d\xi-\mathds{1}_{\Omega_{N}^{\prime}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}\frac{m_{fc}^{\prime}}{N}\sum(g_{i}-{\mathbb{E}}g_{i})f(\xi)d\xi\Big{|}\leq
C_{3}N^{-c_{0}}\sup_{\xi\in\Gamma}|f(\xi)|$

for some constant $C_{3}$. Since both ${\mathbb{P}}(\Omega_{N})$ and
${\mathbb{P}}(\Omega_{N})$ go to 1, we see that

$\mathds{1}_{\Omega_{N}^{\prime}}\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}(m_{fc}(\xi)-\hat{m}_{fc}(\xi))f(\xi)d\xi$

has the same limit in distribution as

$\frac{\sqrt{N}}{2\pi{\text{i}}}\oint_{\Gamma}\frac{m_{fc}^{\prime}}{N}\sum(g_{i}-{\mathbb{E}}g_{i})f(\xi)d\xi=\frac{1}{\sqrt{N}}\sum_{i=1}^{M}\Bigg{\\{}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}m_{fc}^{\prime}(\xi)g_{i}(\xi)f(\xi)d\xi-{\mathbb{E}}\Bigg{[}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}m_{fc}^{\prime}(\xi)g_{i}(\xi)f(\xi)d\xi\Bigg{]}\Bigg{\\}}\\\
=\frac{1}{\sqrt{N}}\sum_{i=1}^{M}\Bigg{\\{}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{\sigma_{i}m_{fc}^{\prime}(\xi)f(\xi)}{1+\sigma_{i}m_{fc(\xi)}}d\xi-{\mathbb{E}}\Bigg{[}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{\sigma_{i}m_{fc}^{\prime}(\xi)f(\xi)}{1+\sigma_{i}m_{fc(\xi)}}d\xi\Bigg{]}\Bigg{\\}}$

which, by central limit theorem, converges in distribution to a centered
Gaussian distribution with variance

${\mathbb{E}}\Bigg{[}\Bigg{(}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{\sigma_{i}m_{fc}^{\prime}(\xi)f(\xi)}{1+\sigma_{i}m_{fc(\xi)}}d\xi\Bigg{)}^{2}\Bigg{]}-\Bigg{(}{\mathbb{E}}\Bigg{[}\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{\sigma_{i}m_{fc}^{\prime}(\xi)f(\xi)}{1+\sigma_{i}m_{fc(\xi)}}d\xi\Bigg{]}\Bigg{)}^{2}\\\
=-\frac{1}{4\pi^{2}}\oint_{\Gamma}\oint_{\Gamma}\int_{l}^{1}\frac{tm_{fc}^{\prime}(\xi_{1})m_{fc}^{\prime}(\xi_{2})f(\xi_{1})f(\xi_{2})}{(1+tm_{fc}(\xi_{1}))(1+tm_{fc}(\xi_{2}))}d\nu(t)d\xi_{1}d\xi_{2}+\Bigg{(}\oint_{\Gamma}f(\xi)m_{fc}^{\prime}(\xi)(\xi+\frac{1}{m_{fc}(\xi)})d\xi\Bigg{)}^{2}$

where we used the fact
$\int_{l}^{1}\frac{t}{1+tm_{fc}(\xi)}=-\xi+\frac{1}{m_{fc}(xi)}$ in the last
identity. So the proof of the main theorem is complete.

## References

  * [1] Zhidong Bai and Jack W. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices, Ann. Probab. 32, 2004.
  * [2] Zhidong Bai and Jack W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, Mathematics Monograph Series 2. Science Press, Beijing, 2006.
  * [3] Z. D. Bai, X. Wang and W. Zhou, Functional CLT for sample covariance matrices, Bernoulli 16 (4), 1086–1113 (2010).
  * [4] J. Baik and J. O. Lee, Free energy of bipartite spherical Sherrington–Kirkpatrick model, Ann. Inst. H. Poincaré Probab. Statist. 56, 2020.
  * [5] Florent Benaych-Georges and Antti Knowles, Lectures on the local semicircle law for Wigner matrices, In Advanced Topics in Random Matrices, Panoramas et Synthèses 53, Société Mathématique de France, 2016.
  * [6] A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau and J. Yin, Isotropic local laws for sample covariance and generalized Wigner matrices, Electron. J. Probab. 19, 2014.
  * [7] L. Erdős and H.-T. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, Providence, 2017
  * [8] W. Hachem, A. Hardy, and J. Najim, Large complex correlated Wishart matrices: fluctuations and asymptotic independence at the edges, The Annals of Probability 44 (2016).
  * [9] Hong Chang Ji and Ji Oon Lee, Central limit theorem for linear spectral statistics of deformed Wigner matrices, Random Matrices: Theory Appl. 9, 2020.
  * [10] K. Johansson, Some limit theorems for the eigenvalues of a sample covariance matrix, J. Multivariate Anal. bf 12 (1), 1–38 (1982).
  * [11] Antti Knowles and Jun Yin, Anisotropic local laws for random matrices, Probab. Theory Relat. Fields 169, 2017.
  * [12] J. Kwak, J. O. Lee and J. Park, Extremal eigenvalues of sample covariance matrices with general population, Bernoulli 27, 2021.
  * [13] Ji Oon Lee and Yiting Li, Spherical Sherrington-Kirkpatrick model for deformed Wigner matrix with fast decaying edges, preprint, arXiv:2112.14107.
  * [14] Ji Oon Lee and Kevin Schnelli, Tracy–Widom distribution for the largest eigenvalues of real sample covariance matrices with general population, The Annals of Applied Probability 26, 2016.
  * [15] Yiting Li, Kevin Schnelli and Yuanyuan Xu, Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021).
  * [16] Zhijun Liu, Jiang Hu, Zhidong Bai, Haiyan Song, A CLT for the LSS of large dimensional sample covariance matrices with unbounded dispersions, arXiv:2205.07280, 2022.
  * [17] A. Lytova and L. Pastur, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, Ann. Prob. 37(5), 1778-1840 (2009).
  * [18] V. A. Marchenko and L. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR, Sb. 1 (4), 457–483 (1967).
  * [19] J. Najim and J. Yao, Gaussian fluctuations for linear spectral statistics of large random covariance matrices, Ann. Appl. Probab. 26 (3), 1837–1887 (2016).
  * [20] G. Pan and W. Zhou, Central limit theorem for signal-to-interference ratio of reduced rank linear receiver, The Annals of Applied Probability 18 (3), 1232–1270 (2008).
  * [21] M. Shcherbina, Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices, Zh. Mat. Fiz. Anal. Geom. 7(2), 176-192 (2011).
  * [22] J. W. Silverstein and S. I. Choi, Analysis of the limiting spectral distribution of large dimensional random matrices, J. Multivariate Anal. 54 (1995).
  * [23] D. Voiculescu, Multiplication of certain non-commuting random variables, J. Operator Theory 18 (2), 223–235 (1987).
  * [24] D. Voiculescu, K. J. Dykema and A. Nica, Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, American Mathematical Society, Providence, 1992.