# Tracy-Widom distribution for the edge eigenvalues of elliptical model

Xiucai Ding E-mail<EMAIL_ADDRESS>The author is partially supported by
NSF-DMS 2113489. Department of Statistics, University of California, Davis
Jiahui Xie E-mail<EMAIL_ADDRESS>Department of Statistics and Data
Science, National University of Singapore

###### Abstract

In this paper, we study the largest eigenvalues of sample covariance matrices
with elliptically distributed data. We consider the sample covariance matrix
$Q=YY^{*},$ where the data matrix $Y\in\mathbb{R}^{p\times n}$ contains i.i.d.
$p$-dimensional observations
$\mathbf{y}_{i}=\xi_{i}T\mathbf{u}_{i},\;i=1,\dots,n.$ Here $\mathbf{u}_{i}$
is distributed on the unit sphere, $\xi_{i}\sim\xi$ is independent of
$\mathbf{u}_{i}$ and $T^{*}T=\Sigma$ is some deterministic matrix. Under some
mild regularity assumptions of $\Sigma,$ assuming $\xi^{2}$ has bounded
support and certain proper behavior near its edge so that the limiting
spectral distribution (LSD) of $Q$ has a square decay behavior near the
spectral edge, we prove that the Tracy-Widom law holds for the largest
eigenvalues of $Q$ when $p$ and $n$ are comparably large.

##  1 Introduction

Large dimensional sample covariance matrices play important roles in modern
statistical learning theory. Understanding the behavior of the largest
eigenvalues of the sample covariance matrices is crucial in many statistical
techniques, especially the principal component analysis (PCA). Consider
$\mathbf{y}_{i}\in\mathbb{R}^{p},1\leqslant i\leqslant n,$ be a sequence of
i.i.d. mean zero random vectors, after properly being scaled, the associated
sample covariance matrix can be written as

$Q=\sum_{i=1}^{n}\mathbf{y}_{i}\mathbf{y}_{i}^{*}.$

Lots of efforts have been made when
$\mathbf{y}_{i}=\Sigma^{1/2}\mathbf{x}_{i},$ where $\Sigma$ is some positive
define matrix representing the population covariance structures and
$\mathbf{x}_{i}$ contains i.i.d. mean zero entries with variance $n^{-1}.$ In
summary, under certain regularity assumptions on $\Sigma$ and moment
conditions on $\mathbf{x}_{i},$ when $p$ and $n$ are comparably large, after
being properly scaled and centered, the largest eigenvalue of $Q$ follows
Tracy-Widom (TW) distribution asymptotically. To list but a few, see [1, 4, 6,
7, 11, 15, 19].

In this work, we extend this line of research to the setting when
$\mathbf{y}_{i}$ follows elliptical distribution [9]

$\mathbf{y}_{i}=\xi_{i}T\mathbf{u}_{i}\in\mathbb{R}^{p},$ (1.1)

where $\xi_{i}\in\mathbb{R}$ are i.i.d. random variables,
$T^{*}T=\Sigma\in\mathbb{R}^{p\times p}$ is some positive definite
deterministic matrix, and $\mathbf{u}_{i}\in\mathbb{R}^{p},1\leqslant
i\leqslant n,$ are distributed on the unit sphere
$\mathrm{U}(\mathbb{S}^{p-1})$ that is independent of $\\{\xi_{i}\\}$. We can
write the data matrix $Y=TUD$, where $U=(\mathbf{u}_{i})$ and $D$ is a
diagonal matrix containing $\\{\xi_{i}\\}$ satisfying certain regularity
assumptions; see Assumption 2.1. Then the sample covariance matrix $Q$ can be
rewritten as

$\displaystyle Q=TUD^{2}U^{*}T^{*}.$ (1.2)

We consider the high dimensional setting that for some small fixed constant
$0<\tau<1$

$\tau\leqslant\phi:=\frac{p}{n}\leqslant\tau^{-1}.$ (1.3)

It has been shown in [8, 10], the empirical spectral distribution (ESD) of $Q$
can be best formulated by its Stieltjes transform which can be described via a
system of two equations. Much less is known about the individual eigenvalues
except a recent one [20]. However, [20] requires that $\\{\xi_{i}^{2}\\}$ are
infinitely divisible so that $D^{2}$ is almost deterministic. As a
consequence, the system of two equations will degenerate to only one. In the
current paper, we consider a more challenging setting that $\xi_{i}^{2}$ are
truly random so that the limiting ESD will be governed by two equations. Under
mild conditions, we prove that the largest eigenvalue of $Q$ follows Tracy-
Widom distribution asymptotically, after being properly centered and scaled;
see Theorem 2.8.

This paper is organized as follows. In Section 2, we provide some useful
definitions and state the main results. In Section 3, we provide some
preliminary results. In Section 4, we prove the main results. Finally, the
proof of local laws is sketched in Appendix A.

##  2 Definitions and main results

###  2.1 The model and asymptotic laws

Throughout the paper, we consider the observations (1.1) and their associated
sample covariance matrix (1.2). Due to rotational invariance, we assume that
$\Sigma$ is a diagonal matrix so that
$\Sigma=\operatorname{diag}\left\\{\sigma_{1},\cdots,\sigma_{p}\right\\}.$
Moreover, for some small constant $0<\tau<1,$ we assume that

$\tau\leqslant\sigma_{p}\leqslant\sigma_{p-1}\leqslant\cdots\leqslant\sigma_{2}\leqslant\sigma_{1}\leqslant\tau^{-1}.$
(2.1)

For the diagonal matrix
$D^{2}=\operatorname{diag}\left\\{\xi_{1}^{2},\cdots,\xi_{n}^{2}\right\\}$ in
(1.2) (or equivalently the random variables $\xi_{i},1\leqslant i\leqslant n,$
in (1.1)), we impose the following assumptions. Similar conditions have been
used in [12, 13, 14, 16].

######  Assumption 2.1.

We assume $\xi_{i}^{2}\sim\xi^{2},1\leqslant i\leqslant n,$ are i.i.d. random
variables. Moreover, we assume that $\xi^{2}$ has a bounded support on $(0,l]$
for fixed some constant $l>0.$ Moreover, for some constant $d>-1,$ we assume
that

$\mathbb{P}(l-\xi^{2}\leqslant x)\asymp x^{d+1}.$ (2.2)

Then we prepare some notations. For the sample covariance matrix $Q$ in (1.2)
and its companion $\mathcal{Q},$

$\mathcal{Q}:=DU^{*}T^{*}TUD\equiv DU^{*}\Sigma UD,$ (2.3)

their empirical spectral distributions (ESD) are defined as
$\mu_{Q}:=p^{-1}\sum_{i=1}^{p}\delta_{\lambda_{i}(Q)}$ and
$\mu_{\mathcal{Q}}:=n^{-1}\sum_{j=1}^{n}\delta_{\lambda_{j}(\mathcal{Q})},$
respectively. Correspondingly, the Stieltjes transforms are denoted as

$m_{Q}:=\int\frac{1}{x-z}\mu_{Q},\quad
m_{\mathcal{Q}}:=\int\frac{1}{x-z}\mu_{\mathcal{Q}},\ z\in\mathbb{C}_{+}.$
(2.4)

Since $Q$ and $\mathcal{Q}$ share the same non-trivial eigenvalues, it
suffices to study $\mu_{Q}.$ To characterize the limit of $\mu_{Q},$ we
consider a system of equations.

######  Definition 2.2 (System of consistent equations).

For $z\in\mathbb{C}_{+},$ we define the triplets
$(m_{1n},m_{2n},m_{n})\in\mathbb{C}^{3}_{+},$ via the following system of
equations.

$\displaystyle
m_{1n}(z)=\frac{1}{p}\sum_{i=1}^{p}\frac{\sigma_{i}}{-z(1+\sigma_{i}m_{2n}(z))},\quad
m_{2n}(z)=\frac{1}{p}\sum_{i=1}^{n}\frac{\xi_{i}^{2}}{-z(1+\xi^{2}_{i}m_{1n}(z))}$
(2.5) $\displaystyle
m_{n}(z)=\frac{1}{p}\sum_{i=1}^{p}\frac{1}{-z(1+\sigma_{i}m_{2n}(z))}.$

Moreover, we define the the triplets
$(m_{1n,c},m_{2n,c},m_{n,c})\in\mathbb{C}^{3}_{+},$ via the following system
of equations.

$\displaystyle
m_{1n,c}(z)=\frac{1}{p}\sum_{i=1}^{p}\frac{\sigma_{i}}{-z(1+\sigma_{i}m_{2n,c}(z))},\quad
m_{2n,c}(z)=\phi^{-1}\int_{0}^{l}\frac{s}{-z(1+sm_{1n,c}(z))}\mathrm{d}F(s)$
(2.6) $\displaystyle
m_{n,c}(z)=\frac{1}{p}\sum_{i=1}^{p}\frac{1}{-z(1+\sigma_{i}m_{2n,c}(z))},$

where $F(s)$ is the cumulative distribution (CDF) of $\xi^{2}$ and recall
$\phi$ in (1.3).

We point out that according to Lemma 3.1 below, (2.6) can be regarded as an
asymptotic deterministic equivalent of (2.5). To avoid repetitions, we
summarize some assumptions as follows.

######  Assumption 2.3.

We assume (1.1), (1.3), (2.1) and Assumption 2.1 hold.

######  Theorem 2.4.

Suppose Assumption 2.3 holds. Then conditional on some event
$\Omega\equiv\Omega_{n}$ (defined in Lemma 3.1) that
$\mathbb{P}(\Omega)=1-\mathrm{o}(1),$ for any $z\in\mathbb{C}_{+},$ when $n$
is sufficiently large, there exists a unique solution
$(m_{1n}(z),m_{2n}(z),m_{n}(z))\in\mathbb{C}_{+}^{3}$ to the system of
equations in (2.5). Moreover, $m_{n}(z)$ is the Stieltjes transform of some
probability measure $\mu\equiv\mu_{n}$ defined on $\mathbb{R}$ which can be
obtained using the inversion formula and has a continuous derivative $\rho$ on
$(0,\infty)$. Similar results hold for $(m_{1n,c},m_{2n,c},m_{n,c})$ in (2.6)
unconditionally.

###### Proof.

The proofs can be obtained by following lines of the arguments of [8, Theorem
2] and [18, Theorem 1], or see [5, Theorem 2.4] verbatim. We omit the details.
∎

Thanks to Theorem 2.4, it is easy to see that the study of the system of
equations, for example (2.5), can be reduced to the analysis of

$F_{p}(m_{1n}(z),z)=0,\quad z\in\mathbb{C}_{+},$ (2.7)

where $F_{p}(\cdot,\cdot)$ are defined as follows

$F_{p}(m_{1n}(z),z)=\frac{1}{p}\sum_{i=1}^{p}\frac{\sigma_{i}}{-z+\frac{\sigma_{i}}{p}\sum_{j=1}^{n}\frac{\xi^{2}_{j}}{1+\xi^{2}_{j}m_{1n}(z)}}-m_{1n}(z).$
(2.8)

Moreover, the right-most edge of the support can also be characterized using
(2.8). Denote $\rho,\rho_{1},\rho_{2},\rho_{c},\rho_{1c}$ and $\rho_{2c}$ as
the density functions associated with the Stieltjes transforms
$m_{n},m_{1n},m_{2n},m_{n,c},m_{1n,c}$ and $m_{2n,c}$ in Theorem 2.4 via the
inversion formula. Let $\lambda_{+}$ be the right-most edge of the support of
$\rho$ and $L_{+}$ be that of $\rho_{c}.$ Then we have that

######  Lemma 2.5.

Suppose the assumptions of Theorem 2.4 hold. Then condition on the event
$\Omega,$ we have that

$\displaystyle\operatorname{supp}\rho\cap(0,\infty)=\operatorname{supp}\rho_{1(2)}\cap(0,\infty).$

Moreover, $(x,y)=(m_{1n}(\lambda_{+}),\lambda_{+})$ is the real solution that
satisfies the following equations

$\displaystyle F_{p}(x,y)=0,\quad\frac{\partial F_{p}}{\partial x}(x,y)=0.$
(2.9)

Similar results hold for $\rho_{c},\rho_{1(2),c}$ and $L_{+}$ unconditionally
when $F_{p}$ is replaced by $F_{p,c}$ defined using (2.6) that

$F_{p,c}(x,y)=\frac{1}{p}\sum_{i=1}^{p}\frac{\sigma_{i}}{-y+\phi^{-1}\sigma_{i}\int\frac{s}{1+sx}\mathrm{d}F(s)}-x.$
(2.10)

###### Proof.

See Lemma 2.5 of [21]. ∎

To study the edge behaviors, we need the following assumption which guarantees
a regular square-root behavior of the spectral densities near the edges.

######  Assumption 2.6.

Suppose that for some constant $\tau>0,$

$\displaystyle|1+\sigma_{1}m_{2n,c}(L_{+})|\geq\tau.$ (2.11)

Moreover, for $\xi^{2}$ in Assumption 2.1, we assume that either (1).
$-1<d\leqslant 1$ or (2). $d>1$ and $\phi^{-1}<\vartheta$, where
$\vartheta\equiv\vartheta(\xi^{2})$ is defined as

$\vartheta=\frac{\phi^{-1}}{p}\sum_{i=1}^{p}\frac{\sigma_{i}^{2}\upsilon_{1}}{(L_{+}-\sigma_{i}\upsilon_{2})^{2}},\
\text{where}\
\upsilon_{1}:=\phi^{-1}\int_{0}^{l}\frac{l^{2}s^{2}}{(l-s)^{2}}\mathrm{d}F(s)\
\text{and}\ \upsilon_{2}:=\phi^{-1}\int_{0}^{l}\frac{ls}{l-s}\mathrm{d}F(s).$

######  Remark 2.7.

As will be seen in Theorem 3.2 below, the conditions in Assumption 2.6 ensure
that $\rho_{c},\rho_{1,c},\rho_{2,c}$ have a square root decay behavior near
$L_{+}.$ Furthermore, it also implies that $\rho,\rho_{1},\rho_{2}$ have a
square root decay behavior near $\lambda_{+}$ on $\Omega.$ We remark that the
square root behavior of the LSD is generally believed to be necessary
condition for the appearance of the Tracy-Widom law in the asymptotic limit.
For example, if the LSD has a linear behavior near the edge, then the
corresponding asymptotics will be very different [3, 12, 13, 14].

###  2.2 Main results

The main theorem of the paper is stated as follows. Let
$\gamma_{0}\equiv\gamma_{0}(n)$ be defined as follows

$\gamma_{0}^{3}=\frac{-2\partial_{y}F_{p,c}(m_{1n,c}(L_{+}),L_{+})}{\partial^{2}_{x}F_{p,c}(m_{1n,c}(L_{+}),L_{+})}\left(\phi^{-1}\int_{0}^{l}\frac{s}{L_{+}(1+sm_{1n,c}(L_{+}))^{2}}\mathrm{d}F(s)\right)^{2}.$
(2.12)

Let $\lambda_{1}\geqslant\lambda_{2}\geqslant\lambda_{\min\\{p,n\\}}>0$ be the
non-zero eigenvalues of $Q.$

######  Theorem 2.8.

Suppose Assumptions 2.3 and 2.6 hold. Then we have that for all
$x\in\mathbb{R}$

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\big{(}\gamma_{0}n^{2/3}(\lambda_{1}-\lambda_{+})\leq
x\big{)}=F_{1}(x),$ (2.13)

where $\lambda_{+}$ is defined in Lemma 2.5 via (2.9), $\gamma_{0}$ is defined
in (2.12) and $F_{1}$ is the type-1 Tracy-Widom cumulative distribution
function.

######  Remark 2.9.

Several remarks are in order. First, in (2.13), as discussed in Remark 2.9 of
[21], $\gamma_{0}$ is deterministic and $\gamma_{0}\asymp 1$. In addition,
$\lambda_{+}$ is random in general. Second, similar to Theorem III.2 of [6],
we can generalize (2.13) to multiple edge eigenvalues. Recall the Gaussian
orthogonal ensemble (GOE) refers to symmetric random matrices of the form
$(X+X^{*})/\sqrt{2},$ where $X$ is a $n\times n$ matrix with i.i.d. Gaussian
entries with mean zero and variance $n^{-1}.$ Let $\\{\mu_{i}^{\text{GOE}}\\}$
be the eigenvalues of GOE in the decreasing order, then for any fixed
$k\in\mathbb{N},$ we can prove that for all
$(x_{1},x_{2},\cdots,x_{k})\in\mathbb{R}^{k}$

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\left[\left(\gamma_{0}n^{2/3}(\lambda_{i}-\lambda_{+})\leq
x_{i}\right)_{1\leqslant i\leqslant
k}\right]=\lim_{n\rightarrow\infty}\mathbb{P}\left[\left(n^{2/3}(\mu^{\text{GOE}}_{i}-2)\leq
x_{i}\right)_{1\leqslant i\leqslant k}\right].$

Third, we compare our results with two related works. On the one hand, in [3],
the authors also consider the edge eigenvalues of (1.2) with a focus that the
LSD can be either unbounded or have a linear decay behavior near the edge.
Therefore, the asymptotics are not Tracy-Widom in general. On the other hand,
in [20], the authors assumed that $\xi^{2}$ had an infinite divisible
representation so that the system of equations in Definition 2.2 reduces to
only one. As a result, $\xi_{i}$’s in (1.1) are almost deterministic so that
the proof follows straightforwardly from those of the sample covariance matrix
with i.i.d. entries or Wigner matrices, for example, see [1, 4, 17, 19].

The proof of Theorem 2.8 relies on the Green function comparison approach as
in [1, 4, 16, 19], except that we will conduct an expansion up to the order of
four as in [2]. In Corollary 1 of [6], it was shown that when $U$ in (1.2) is
replaced by a Gaussian matrix, its largest eigenvalue will follow TW law
asymptotically. More specifically, conditional on some realization of $D$
which happens with high probability (c.f. Lemma 3.1), consider
$Q^{G}=TZD^{2}Z^{*}T^{*},$ where $Z=(\mathbf{z}_{i})$ and
$\mathbf{z}_{i}\overset{\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,p^{-1}I_{p\times
p}),$ it has been proved that
$\gamma_{0}n^{2/3}(\lambda_{1}(Q^{G})-\lambda_{+})$ follows TW law
asymptotically. Based on this, it suffices to conduct the comparison argument
between $Q$ and $Q^{G}$ which relies on two important ingredients. First, in
Lemma 3.2, we conduct the local analysis for the system of equations in
Definition 2.2 from which we observe the square root behavior near the edge.
Second, for comparison, in Theorem 3.4, we prove the local laws for the
following matrices

$\displaystyle S:=VV^{*}\equiv
TWD^{2}W^{*}T^{*},\quad\mathcal{S}:=V^{*}V\equiv DW^{*}\Sigma WD,\ \ V:=TWD,$
(2.14)

where $W=(\mathbf{w}_{i})$ with
$\mathbf{w}_{i}=\mathbf{u}_{i}\;\text{or}\;\mathbf{z}_{i}.$ We note that if
$W=U$ or $W=Z$, then $S\equiv Q$ or $S\equiv Q^{G}$, respectively.

##  3 Preliminaries

We provide some preliminary results. First, since $D^{2}$ is random, it is
more convenient to fixed some realization with certain properties which
happens with high probability. The following lemma provides such probability
events.

######  Lemma 3.1.

Under Assumptions 2.1, denote $\Omega\equiv\Omega_{n}$ be the event on
$\\{\xi_{i}^{2}\\}$ so that the following conditions hold:

$\displaystyle n^{-1/(d+1)}\log^{-1}n<l-\xi^{2}_{(1)}<n^{-1/(d+1)}\log n,\ \
\xi^{2}_{(1)}-\xi^{2}_{(2)}>n^{-1/(d+1)}\log^{-1}n,$ (3.1)
$\displaystyle\left|\frac{1}{n}\sum_{i=1}^{n}\frac{\xi^{2}_{i}}{1+\xi^{2}_{i}m_{1n}(z)}-\int\frac{t}{1+tm_{1n}(z)}\mathrm{d}F(t)\right|\leq\frac{Cn^{\epsilon}}{\sqrt{n}},$

where $C>0$ is some generic constant and $\epsilon>0$ is some arbitrarily
small constant. Then we have that $\mathbb{P}(\Omega)=1-\mathrm{o}(1).$

###### Proof.

See Lemma A.11 of [3]. ∎

Second, we summarize the properties of the Stieltjes transforms. For some
fixed (small) constants $\mathsf{c},\mathsf{C},\epsilon_{e}>0,$ define the
spectral parameter sets

$\mathbf{D}\equiv\mathbf{D}(\mathtt{c},\mathtt{C},\varepsilon_{e}):=\left\\{z=E+\mathrm{i}\eta\in\mathbb{C}_{+}:\lambda_{+}-\mathtt{c}\leq
E\leq\lambda_{+}+\mathtt{C},\;p^{-1+\varepsilon_{e}}\leq\eta\leq\mathtt{C}\right\\},$
(3.2)

and

$\mathbf{D}_{0}\equiv\mathbf{D}(\mathtt{c},\mathtt{C},\infty):=\left\\{z=E+\mathrm{i}\eta\in\mathbb{C}_{+}:\lambda_{+}-\mathtt{c}\leq
E\leq\lambda_{+}+\mathtt{C},\;0<\eta\leq\mathtt{C}\right\\}.$ (3.3)

######  Lemma 3.2.

Suppose Assumptions 2.3 and 2.6 hold. When $n$ is sufficiently large, for any
realization $\\{\xi^{2}_{i}\\}\in\Omega$ defined in Lemma 3.1, we have that

  * (1)

For all $z\in\mathbf{D}$ with $z\rightarrow\lambda_{+}$,

$\displaystyle m_{n}(\lambda_{+})-m_{n}(z)\asymp\sqrt{|\lambda_{+}-z|}.$ (3.4)

Consequently, for small constant $\kappa>0$

$\displaystyle\rho(\lambda_{+}-\kappa)\asymp\sqrt{\kappa}.$ (3.5)

The results also apply to $m_{1(2)n}$ and $\rho_{1(2)}.$

  * (2)

For all $z\in\mathbf{D}_{0}$,

$\displaystyle|m_{n}(z)|\sim
1,\quad\operatorname{Im}m_{n}(z)\sim\begin{cases}\eta/\sqrt{\kappa+\eta},&\quad\operatorname{if}\quad
E\geq\lambda_{+}\\\ \sqrt{\kappa+\eta},&\quad\operatorname{if}\quad
E\leq\lambda_{+}.\end{cases}$ (3.6)

The results also apply to $m_{1(2)n}.$

  * (3)

There exists an constant $\tau^{\prime}>0$, such that for any
$z\in\mathbf{D}_{0}$

$\displaystyle\min_{1\leqslant i\leqslant
p}|1+\sigma_{i}m_{2n}(z)|\geq\tau^{\prime},\quad\min_{1\leqslant j\leqslant
n}|1+\xi^{2}_{j}m_{1n}(z)|\geq\tau^{\prime}.$ (3.7)

Finally, all the results in (3.4)–(3.7) also hold for
$m_{n,c},m_{1(2)n,c},\rho_{c},\rho_{1(2)c}$ unconditionally without fixing the
realization.

###### Proof.

See Lemma A.4 of [3]. ∎

Third, we provide the local laws. The following notion will be used in the
statements.

######  Definition 3.3 (Stochastic domination).

Let

$A=\left(A^{(n)}(u):n\in\mathbb{N},u\in U^{(n)}\right),\hskip
10.0ptB=\left(B^{(n)}(u):n\in\mathbb{N},u\in U^{(n)}\right),$

be two families of nonnegative random variables, where $U^{(n)}$ is a possibly
$n$-dependent parameter set. We say $A$ is stochastically dominated by $B$,
uniformly in $u$, if for any fixed (small) $\epsilon>0$ and (large) $\xi>0$,

$\sup_{u\in
U^{(n)}}\mathbb{P}\left(A^{(n)}(u)>n^{\epsilon}B^{(n)}(u)\right)\leq
n^{-\xi},$

for large enough $n\geq n_{0}(\epsilon,\xi)$, and we shall use the notation
$A\prec B$ or $A=\mathrm{O}_{\prec}(B)$. Throughout this paper, the stochastic
domination will always be uniform in all parameters that are not explicitly
fixed, such as the matrix indices and the spectral parameter $z$.

Recall (2.14). For $z=E+\mathrm{i}\eta\in\mathbb{C}_{+},$ denote the
resolvents

$G(z)=(S-zI)^{-1}\in\mathbb{R}^{p\times
p},\quad\mathcal{G}(z)=(\mathcal{S}-zI)^{-1}\in\mathbb{R}^{n\times n}.$ (3.8)

Moreover, we define $m(z):=p^{-1}{\rm tr}(G(z))$ and
$m_{1}(z):=p^{-1}\operatorname{tr}(G(z)\Sigma).$

######  Theorem 3.4.

Suppose Assumptions 2.3 and 2.6 hold. When $n$ is sufficiently large, for any
realization $\\{\xi^{2}_{i}\\}\in\Omega$ defined in Lemma 3.1, we have that
for $z\in\mathbf{D}$ in (3.2) uniformly,

$\displaystyle\max_{1\leqslant i,j\leqslant
n}|\mathcal{G}+z^{-1}(1+m_{1n}(z)D^{2})^{-1}|_{ij}\prec\sqrt{\frac{\operatorname{Im}m_{1n}(z)}{p\eta}}+\frac{1}{p\eta}.$
(3.9)

Moreover,

$\displaystyle{|m_{1n}(z)-m_{1}(z)|+|m_{n}(z)-m(z)|\prec(p\eta)^{-1}.}$ (3.10)

Finally, for any $z\in\mathbf{D}\cap\\{z=E+\mathrm{i}\eta:E\geq\lambda_{+},\
p\eta\sqrt{\kappa+\eta}\geq p^{\varepsilon_{e}}\\}$ uniformly,

$\displaystyle{|m_{1n}(z)-m_{1}(z)|+|m_{n}(z)-m(z)|\prec\frac{1}{p(\kappa+\eta)}+\frac{1}{(p\eta)^{2}\sqrt{\kappa+\eta}}.}$
(3.11)

##  4 Main technical proofs

As discussed in the end of Section 2.2, our results rely on a comparison
argument. The core is to prove the following argument. Let
$\widetilde{G}(z):=(Q^{G}-zI)^{-1}$ be resolvent of $Q^{G}$ and
$\widetilde{m}(z):=p^{-1}\operatorname{tr}\widetilde{G}(z)$.

######  Proposition 4.1.

Suppose the assumptions of Lemma 3.2 hold and
$F:\mathbb{R}\rightarrow\mathbb{R}$ is a function whose derivatives satisfy

$\sup_{x\in\mathbb{R}}|F^{(l)}(x)|(1+|x|)^{-C_{1}}\leq C_{1},\qquad
l=1,2,3,4,$

for some constant $C_{1}>0$. Then for any sufficiently small constant
$\epsilon>0$ and for any real numbers $E,E_{1}$ and $E_{2}$ satisfying

$|E-\lambda_{+}|\leq n^{-2/3+\epsilon},\ |E_{1}-\lambda_{+}|\leq
n^{-2/3+\epsilon},\ |E_{2}-\lambda_{+}|\leq n^{-2/3+\epsilon},$ (4.1)

and $\eta_{0}=n^{-2/3-\epsilon}$, there exist some constants $c_{1},C>0$

$\displaystyle|\mathbb{E}F(n\eta_{0}\operatorname{Im}m(z))-\mathbb{E}F(n\eta_{0}\operatorname{Im}{\widetilde{m}(z))}|\leq
n^{-c_{1}+C{\epsilon}},\qquad z=E+\mathrm{i}\eta_{0},$ (4.2)

and

$\displaystyle\Big{|}\mathbb{E}F\Big{(}\int_{E_{1}}^{E_{2}}n\operatorname{Im}m(y+\mathrm{i}\eta_{0}){\rm
d}y\Big{)}-\mathbb{E}F\Big{(}\int_{E_{1}}^{E_{2}}n\operatorname{Im}{\widetilde{m}(y+\mathrm{i}\eta_{0})}{\rm
d}y\Big{)}\Big{|}\leq n^{-c_{1}+C{\epsilon}}.$ (4.3)

Armed with Proposition 4.1, we are able to prove the main result, Theorem 2.8.

###### Proof of Theorem 2.8.

Using Proposition 4.1, following lines of the arguments of Corollary 4.2 of
[19] or Lemmas 5.1 and 5.2 of [4], we can show that for any fixed realization
$\\{\xi_{i}^{2}\\}\in\Omega$ in Lemma 3.1 and any $x\in\mathbb{R},$

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}(n^{2/3}(\lambda_{1}(Q)-\lambda_{+})\leq
x)=\lim_{n\rightarrow\infty}\mathbb{P}(n^{2/3}(\lambda_{1}(Q^{G})-\lambda_{+})\leq
x).$ (4.4)

It has been proved in Corollary 1 of [6] and Section 2.3 [21] that

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}(\gamma_{0}n^{2/3}(\lambda_{1}(Q^{G})-\lambda_{+})\leq
x)=F_{1}(x).$ (4.5)

Moreover, as discussed in Remark 2.9, we have that $\gamma_{0}\asymp 1.$ For
the unconditional setting, we see from Lemma 3.1 that
$\mathbb{P}(\Omega)=1-\mathrm{o}(1)$. Therefore, Theorem 2.8 can be proved
using(4.4) and (4.5). ∎

The rest of the section leaves to the proof of Proposition 4.1.

###  4.1 Proof of Proposition 4.1

We first prepare some notations. For the data matrix $V$ in (2.14) and the
index set ${\cal I}=\\{1,\dots,n\\}$, given $\mathcal{T}\subset{\cal I}$, we
introduce the notation $V^{(\mathcal{T})}$ to denote the
$p\times(n-|\mathcal{T}|)$ minor of $V$ obtained from removing all the $i$th
columns of $V$ for $i\in\mathcal{T}$ and keep the original indices of $V$. For
convenience, we briefly write $\\{i\\}$, $\\{i,j\\}$ and
$\\{i,j\\}\cup\mathcal{T}$ as $(i)$, $(ij)$ and $(ij\mathcal{T})$
respectively. Correspondingly, we denote the matrices
$S^{(\mathcal{T})}=(V^{(\mathcal{T})})(V^{(\mathcal{T})})^{*},\
{\mathcal{S}}^{(\mathcal{T})}=(V^{(\mathcal{T})})^{*}(V^{(\mathcal{T})})$ and
their associated resolvents as
$G^{(\mathcal{T})}(z)=(S^{(\mathcal{T})}-zI)^{-1}$ and ${\cal
G}^{(\mathcal{T})}(z)=({\mathcal{S}}^{(\mathcal{T})}-zI)^{-1},$ respectively.

###### Proof of Proposition 4.1.

Due to similarity, we only prove (4.2). We consider the following Linderberg
replacement for $U$. For $\gamma=0,\dots,n$, let $W_{\gamma}$ be the matrix
whose first $\gamma$ columns are the same as those of $Z$ and the remaining
$n-\gamma$ columns are the same as those of $U$. Then, it is easy to see
$W_{0}=U$ and $W_{n}=Z$. Denote $G_{\gamma}$, $\mathcal{G}_{\gamma}$ as the
Green functions of $TW_{\gamma}D^{2}W_{\gamma}^{*}T^{*}$ and
$DW_{\gamma}^{*}T^{*}TW_{\gamma}D$ respectively, and
$m_{p,\gamma}=p^{-1}\operatorname{tr}G_{\gamma}$,
$m_{n,\gamma}=p^{-1}\operatorname{tr}\mathcal{G}_{\gamma}$. We now rewrite
(4.2) as the following telescoping summation

$\displaystyle\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m(z))\big{)}-\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}\widetilde{m}(z))\big{)}=\sum_{\gamma=1}^{n}\Big{(}\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{p,\gamma-1}(z))\big{)}-\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{p,\gamma}(z))\big{)}\Big{)}.$

It suffices to prove that for all $1\leqslant\gamma\leqslant n$

$\displaystyle|\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{p,\gamma-1}(z))\big{)}-\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{p,\gamma}(z))\big{)}|\leq
n^{-1-c+C\epsilon}.$ (4.6)

Following [1, 4, 19], by introducing
$m_{*,\gamma}=p^{-1}\operatorname{tr}G_{\gamma}^{(\gamma)}$, it suffices to
prove

$\displaystyle\begin{split}\Big{|}\mathbb{E}\big{[}F(n\eta_{0}\operatorname{Im}m_{p,\gamma-1}(z))\big{]}-\mathbb{E}\big{[}F(n\eta_{0}\operatorname{Im}m_{*,\gamma}(z))\big{]}-\Big{(}\mathbb{E}\big{[}F(n\eta_{0}\operatorname{Im}m_{p,\gamma}(z))\big{]}-\mathbb{E}\big{[}F(n\eta_{0}\operatorname{Im}m_{*,\gamma}(z))\big{]}\Big{)}\Big{|}\leq
n^{-1-c_{1}+C\epsilon}.\end{split}$ (4.7)

Throughout the proof, for notational convenience, in (2.14), we set $Y=TUD$ if
$W=U$ and $Y^{G}=TZD$ if $W=Z.$ Moreover, we denote the $\gamma$-th column of
$Y$ and $Y^{G}$ as $\mathbf{y}_{\gamma}$ and $\widetilde{\mathbf{y}}_{\gamma}$
respectively. Our arguments rely on the following estimates whose proof will
be given in Section 4.2.

######  Lemma 4.2.

For $\epsilon$ in (4.1), we have that

$\displaystyle|\mathbf{y}^{*}_{\gamma}(G_{\gamma}^{(\gamma)})^{2}\mathbf{y}_{\gamma}|\prec
n^{1/3+\epsilon},$ (4.8)
$\displaystyle{{|[G_{\gamma}^{(\gamma)}]_{ii}+z^{-1}(1+\sigma_{i}m_{2n}(z))^{-1}|}\prec
n^{-1/3+\epsilon},}\quad|[G_{\gamma}^{(\gamma)}]_{ij}|\prec
n^{-1/3+\epsilon},\;1\leqslant i\neq j\leqslant p,$ (4.9)
$\displaystyle|[(G_{\gamma}^{(\gamma)})^{2}]_{ij}|\prec n^{1/3+2\epsilon},\
1\leqslant i\neq j\leqslant p.$ (4.10)

Set
$\mathcal{X}:=-\mathbf{y}_{\gamma}^{*}(G^{(\gamma)}_{\gamma})^{2}\mathbf{y}_{\gamma}$
and
$\mathcal{Y}:=-\mathbf{y}_{\gamma}^{*}G^{(\gamma)}_{\gamma}\mathbf{y}_{\gamma}$
with their counterparts
$\widetilde{\mathcal{X}}:=-\widetilde{\mathbf{y}}_{\gamma}^{*}(G^{(\gamma)}_{\gamma})^{2}\widetilde{\mathbf{y}}_{\gamma}$
and
$\widetilde{\mathcal{Y}}:=-\widetilde{\mathbf{y}}_{\gamma}^{*}G_{\gamma}^{(\gamma)}\widetilde{\mathbf{y}}_{\gamma}$.
Using Lemma A.3, Theorem 3.4, Lemmas A.2 and 4.2, we have that

$|\mathcal{X}(\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z))^{s}|\prec
n^{-(s-1)/3+(s+1)\epsilon}.$ (4.11)

Using (4.11), (3.6) and decomposing
$\mathcal{Y}=\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z)-\xi^{2}_{\gamma}m_{1n}(z)$,
we conclude from Binomial theorem that for some constants
$C_{s}=\mathrm{O}(1),0\leqslant s\leqslant 2$

$\displaystyle\begin{split}\operatorname{Im}\big{(}\sum_{k\geq
0}\mathcal{X}\mathcal{Y}^{k}\big{)}&=\operatorname{Im}\big{(}\sum_{0\leq s\leq
2}C_{s}\mathcal{X}(\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z))^{s}\big{)}+\operatorname{Im}\big{(}\sum_{l\geq
0}(-\xi^{2}_{\gamma}m_{1n}(z))^{l}\mathcal{X}\big{)}+\mathrm{O}_{\prec}(n^{-2/3+4\epsilon})\\\
&=\operatorname{Im}\big{(}\sum_{0\leq s\leq
2}C_{s}\mathcal{X}(\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z))^{s}\big{)}+\operatorname{Im}\frac{\mathcal{X}}{1+\xi^{2}_{\gamma}m_{1n}(z)}+\mathrm{O}_{\prec}(n^{-2/3+4\epsilon}).\end{split}$
(4.12)

where we used the fact $m_{1n}(\lambda_{+})>-l^{-1}$ as in Lemma A.4 of [3] so
that $|\xi^{2}_{\gamma}m_{1n}(z)|<1$. Since
$G_{\gamma-1}^{(\gamma)}=G_{\gamma}^{(\gamma)},$ we have the following
resolvent expansion

$\displaystyle
G_{\gamma-1}=G_{\gamma}^{(\gamma)}-G_{\gamma-1}\mathbf{y}_{\gamma}\mathbf{y}_{\gamma}^{*}G_{\gamma}^{(\gamma)},$
(4.13)

which results in

$\displaystyle n\eta_{0}|m_{p,\gamma-1}-m_{*,\gamma}|=n\eta_{0}|\sum_{k\geq
0}\mathcal{X}\mathcal{Y}^{k}|=\phi^{-1}\eta_{0}|z[\mathcal{G}_{\gamma-1}]_{\gamma\gamma}\mathbf{y}_{\gamma}^{*}(G_{\gamma}^{(\gamma)})^{2}\mathbf{y}_{\gamma}|.$
(4.14)

Together with Lemma 4.2 and Theorem 3.4, we see that
$n\eta_{0}|m_{p,\gamma-1}-m_{*,\gamma}|\prec n^{-1/3+\epsilon}.$ Using the
above bound and (4.11), combining (4.12), (4.14) and Lemma 4.2, we see that
for some constant $C>0,$ the first difference in (4.7) can be expanded as

$\displaystyle
F(n\eta_{0}\operatorname{Im}m_{p,\gamma-1})-F(n\eta_{0}\operatorname{Im}m_{*,\gamma})$
$\displaystyle=F^{(1)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\phi^{-1}\eta_{0}\operatorname{Im}\big{(}\sum_{0\leq
s\leq
2}C_{s}\mathcal{X}(\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z))^{s}\big{)}+\mathrm{O}_{\prec}(n^{-4/3+3\epsilon})$
$\displaystyle+F^{(2)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\frac{\phi^{-2}\eta^{2}_{0}}{2}\big{[}\operatorname{Im}\big{(}C_{0}\mathcal{X}\big{)}^{2}+2\operatorname{Im}\big{(}C_{0}\mathcal{X}\big{)}\times\operatorname{Im}\big{(}C_{1}\mathcal{X}(\mathcal{Y}+\xi^{2}_{\gamma}m_{1n}(z))\big{)}\big{]}+\mathrm{O}_{\prec}(n^{-4/3+\epsilon})$
$\displaystyle+F^{(3)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\frac{\phi^{-3}\eta^{3}_{0}}{6}\big{[}\operatorname{Im}\big{(}C_{0}\mathcal{X}\big{)}^{3}+\mathrm{O}_{\prec}(n^{-4/3+\epsilon})\big{]}+\mathrm{O}_{\prec}(n^{-4/3+C\epsilon}).$
(4.15)

Moreover, for some constants $C_{l,k}=\mathrm{O}(1),0\leqslant l\leqslant 3$
and $0\leqslant k\leqslant 2,$ using (4.15), we can further obtain

$\displaystyle\begin{split}&\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{p,\gamma-1})\big{)}-\mathbb{E}\big{(}F(n\eta_{0}\operatorname{Im}m_{*,\gamma})\big{)}\\\
&=\mathbb{E}\Big{(}F^{(1)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\phi^{-1}\eta_{0}\operatorname{Im}\big{(}\sum_{0\leq
k\leq
2}C_{1,k}\mathcal{X}\mathcal{Y}^{k}\big{)}\Big{)}+\mathbb{E}\Big{(}F^{(2)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\phi^{-2}\eta^{2}_{0}\big{(}\frac{C_{2,0}}{2}(\operatorname{Im}\mathcal{X})^{2}+C_{2,1}\operatorname{Im}\mathcal{X}\operatorname{Im}\mathcal{X}\mathcal{Y}\big{)}\Big{)}\\\
&+\mathbb{E}\Big{(}F^{(3)}(n\eta_{0}\operatorname{Im}m_{*,\gamma})\times\phi^{-3}\eta^{3}_{0}\big{(}\frac{C_{3,0}}{6}(\operatorname{Im}\mathcal{X})^{3}\big{)})\Big{)}+\mathrm{O}(n^{-4/3+C\epsilon}).\end{split}$

We note that the above expression also holds for the second difference in
(4.7) as we replace $\mathcal{X}$, $\mathcal{Y}$ with
$\mathcal{\widetilde{X}}$, $\mathcal{\widetilde{Y}}$. In light of (4.7), it
suffices to consider the following difference for any $1\leq\alpha\leq 3$,
$1\leq\alpha+\beta\leq 3$,

$\displaystyle\mathbb{E}\Big{(}\eta_{0}^{\alpha}(\mathbf{y}_{\gamma}^{*}(G_{\gamma}^{(\gamma)})^{2}\mathbf{y}_{\gamma})^{\alpha}(\mathbf{y}_{\gamma}^{*}G_{\gamma}^{(\gamma)}\mathbf{y}_{\gamma})^{\beta}\Big{)}-\mathbb{E}\Big{(}\eta_{0}^{\alpha}(\mathbf{\widetilde{y}}_{\gamma}^{*}(G_{\gamma}^{(\gamma)})^{2}\mathbf{\widetilde{y}}_{\gamma})^{\alpha}(\mathbf{\widetilde{y}}_{\gamma}^{*}G_{\gamma}^{(\gamma)}\mathbf{\widetilde{y}}_{\gamma})^{\beta}\Big{)}.$

We only consider the case when $\gamma=1$, and other cases can be handled
similarly.

First, when $\alpha=1$, $\beta=0$, since $\mathbf{y}_{1}$ and
$\mathbf{\widetilde{y}}_{1}$ are independent with $G_{1}^{(1)}$, we conclude
from (A.2) that
$\mathbb{E}\eta_{0}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}-\mathbb{E}\eta_{0}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}=0.$
For $\alpha=3$, $\beta=0$, we have that

$\displaystyle\begin{split}\mathbb{E}\Big{(}\eta_{0}^{3}\big{(}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}\big{)}^{3}\Big{)}-\mathbb{E}\Big{(}\eta_{0}^{3}\big{(}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}\big{)}^{3}\Big{)}\lesssim\eta_{0}^{3}\big{(}\mathcal{M}_{2}+\mathcal{M}^{1}_{4}+\mathcal{M}_{4}^{2}+\mathcal{M}_{6}\big{)}.\end{split}$

where

$\displaystyle\mathcal{M}_{2}:=\mathbb{E}\sum_{i,j,k}\big{(}u_{i1}^{2}u_{j1}^{2}u_{k1}^{2}[(G_{1}^{(1)})^{2}]_{ij}[(G_{1}^{(1)})^{2}]_{jk}[(G_{1}^{(1)})^{2}]_{ki}\big{)}-\mathbb{E}\sum_{i,j,k}\big{(}z_{i1}^{2}z_{j1}^{2}z_{k1}^{2}[(G_{1}^{(1)})^{2}]_{ij}[(G_{1}^{(1)})^{2}]_{jk}[(G_{1}^{(1)})^{2}]_{ki}\big{)},$
$\displaystyle\mathcal{M}^{1}_{4}:=\mathbb{E}\sum_{i,j}\big{(}u_{i1}^{4}u_{j1}^{2}[(G_{1}^{(1)})^{2}]_{ij}^{2}[(G_{1}^{(1)})^{2}]_{ii}\big{)}-\mathbb{E}\sum_{i,j}\big{(}z_{i1}^{4}z_{j1}^{2}[(G_{1}^{(1)})^{2}]_{ij}^{2}[(G_{1}^{(1)})^{2}]_{ii}\big{)},$
$\displaystyle\mathcal{M}^{2}_{4}:=\mathbb{E}\sum_{i,j}\big{(}u_{i1}^{4}u_{j1}^{2}[(G_{1}^{(1)})^{2}]_{ii}^{2}[(G_{1}^{(1)})^{2}]_{jj}\big{)}-\mathbb{E}\sum_{i,j}\big{(}z_{i1}^{4}z_{j1}^{2}[(G_{1}^{(1)})^{2}]_{ii}^{2}[(G_{1}^{(1)})^{2}]_{jj}\big{)},$
$\displaystyle\mathcal{M}_{6}:=\mathbb{E}\sum_{i}u_{i1}^{6}[(G_{1}^{(1)})^{2}]_{ii}^{3}-\mathbb{E}\sum_{i}z_{i1}^{6}[(G_{1}^{(1)})^{2}]_{ii}^{3}.$

It follows from (4.10) and [20, Theorem V.1] that $\mathcal{M}_{2}\lesssim
p^{-1}n^{1+3\epsilon},\quad\mathcal{M}_{4}^{1,2}\lesssim
p^{-1}n^{1+3\epsilon},\quad\mathcal{M}_{6}\lesssim p^{-2}n^{1+3\epsilon}.$
This implies that
$\mathbb{E}\Big{(}\eta_{0}^{3}\big{(}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}\big{)}^{3}\Big{)}-\mathbb{E}\Big{(}\eta_{0}^{3}\big{(}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}\big{)}^{3}\Big{)}\lesssim
n^{-2+3\epsilon}.$

Second, for the other cases, we will need a finer investigation. We focus on
the case $\alpha=1$, $\beta=2$ and the other cases can be handled similarly.
Let $y_{i1}$, $\widetilde{y}_{i1}$ be the $i$-th element of $\mathbf{y}_{1}$
and $\mathbf{\widetilde{y}}_{1}$ respectively. We use the notation
$\mathcal{E}(ab,ij,st):=\mathbb{E}[(G_{1}^{(1)})^{2}]_{ab}[G_{1}^{(1)}]_{ij}[G_{1}^{(1)}]_{st}$
in the sequel. We have

$\displaystyle\begin{split}&\mathbb{E}\big{(}\eta_{0}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}(\mathbf{y}_{1}^{*}G_{1}^{(1)}\mathbf{y}_{1})^{2}\big{)}=\eta_{0}\sum_{k_{1},k_{2},k_{3}}^{*}\mathbb{E}\prod_{i=1}^{3}y_{k_{i}1}^{2}\big{(}\mathcal{E}(k_{1}k_{1},k_{2}k_{2},k_{3}k_{3})+2\mathcal{E}(k_{1}k_{1},k_{2}k_{3},k_{2}k_{3})\\\
&+4\mathcal{E}(k_{1}k_{2},k_{1}k_{2},k_{3}k_{3})+8\mathcal{E}(k_{1}k_{2},k_{1}k_{3},k_{2}k_{3})\big{)}+\eta_{0}\sum_{\mathcal{J}_{6}}\mathbb{E}\prod_{i=1}^{6}y_{k_{i},1}\mathcal{E}(k_{1}k_{2},k_{3}k_{4},k_{5}k_{6}),\end{split}$
(4.16)

where $\sum^{*}_{k_{1},\dots,k_{l}}$ denotes the sum over
$\\{(k_{1},\dots,k_{l})\in\\{1,\dots,p\\}^{l}:\text{$k_{i}$'s are
distinct}\\}$ and $\mathcal{J}_{6}$ denotes the set of indices
$k_{i}\in\\{1,\dots,p\\},i=1,\dots,6$ such that $k_{i}$ appears even number of
times and there is an index $k_{i}$ appears at least four times. Note that the
odd number of an index $k_{i}$ will give the degenerated expectation. This
gives the estimate $\\#\mathcal{J}_{6}\lesssim n^{2}$, together with (4.9) and
(4.10), we have
$|\eta_{0}\sum_{\mathcal{J}_{6}}\mathbb{E}\prod_{i=1}^{6}y_{k_{i}1}\mathcal{E}(k_{1}k_{2},k_{3}k_{4},k_{5}k_{6})|\lesssim{n^{-4/3+C\epsilon}}.$
For other terms in (4.16), we take difference with their counterparts in
$\mathbb{E}(\eta_{0}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}(\mathbf{\widetilde{y}}_{1}^{*}G_{1}^{(1)}\mathbf{\widetilde{y}}_{1})^{2})$

$\displaystyle\mathbb{E}\big{(}\eta_{0}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}(\mathbf{y}_{1}^{*}G_{1}^{(1)}\mathbf{y}_{1})^{2}\big{)}-\mathbb{E}\big{(}\eta_{0}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}(\mathbf{\widetilde{y}}_{1}^{*}G_{1}^{(1)}\mathbf{\widetilde{y}}_{1})^{2}\big{)}=:\sum_{i=1}^{4}2^{i-1}\Delta_{i}+\mathrm{O}(n^{-4/3+C\epsilon}),$

where

$\displaystyle\Delta_{1}:=\eta_{0}\sum_{k_{1},k_{2},k_{3}}^{*}\Delta_{y}\mathcal{E}(k_{1}k_{1},k_{2}k_{2},k_{3}k_{3}),\quad\Delta_{2}:=\eta_{0}\sum_{k_{1},k_{2},k_{3}}^{*}\Delta_{y}\mathcal{E}(k_{1}k_{1},k_{2}k_{3},k_{2}k_{3}),$
$\displaystyle\Delta_{3}:=\eta_{0}\sum_{k_{1},k_{2},k_{3}}^{*}\Delta_{y}\mathcal{E}(k_{1}k_{2},k_{1}k_{2},k_{3}k_{3}),\quad\Delta_{4}:=\eta_{0}\sum_{k_{1},k_{2},k_{3}}^{*}\Delta_{y}\mathcal{E}(k_{1}k_{2},k_{1}k_{3},k_{2}k_{3}),$

with
$\Delta_{y}:=\mathbb{E}(\prod_{i=1}^{3}y_{k_{i}1}^{2}-\prod_{i=1}^{3}\widetilde{y}_{k_{i}1}^{2})$.
In the following, we aim to show that
$\Delta_{i}=\mathrm{O}(n^{-1-c+C\epsilon})$ for each $i$. In fact, one may see
that

$\displaystyle\begin{split}\Delta_{y}=\mathbb{E}\xi^{6}_{1}\big{(}u_{k_{1}1}^{2}u_{k_{2}1}^{2}u_{k_{3}1}^{2}-z_{k_{1}1}^{2}z_{k_{2}1}^{2}z_{k_{3}1}^{2}-3u_{k_{1}1}^{2}u_{k_{2}1}^{2}z_{k_{3}1}^{2}+3u_{k_{1}1}^{2}z_{k_{2}1}^{2}z_{k_{3}1}^{2}\big{)}=\mathrm{O}(n^{-4}),\end{split}$

where we again used [20, Theorem V.1] and the fact that $k_{1},k_{2},k_{3}$
are distinct. Together with Lemma 4.2, we conclude that for each
$i=1,\dots,4$, $\Delta_{i}={\mathrm{O}(n^{-1-c+C\epsilon})}.$ From above
estimates, one can easily obtain that

$\displaystyle\mathbb{E}\big{(}\eta_{0}\mathbf{y}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{y}_{1}(\mathbf{y}_{1}^{*}G_{1}^{(1)}\mathbf{y}_{1})^{2}\big{)}-\mathbb{E}\big{(}\eta_{0}\mathbf{\widetilde{y}}_{1}^{*}(G_{1}^{(1)})^{2}\mathbf{\widetilde{y}}_{1}(\mathbf{\widetilde{y}}_{1}^{*}G_{1}^{(1)}\mathbf{\widetilde{y}}_{1})^{2}\big{)}=\mathrm{O}(n^{-1-c+C\epsilon}).$

Combining the results of all cases of $\alpha$ and $\beta$, we can show (4.7)
and conclude the proof. ∎

###  4.2 Proof of Lemma 4.2

In this subsection, we show the estimates in (4.8)-(4.10) following Appendix B
of [2].

###### Proof of (4.8).

Recall that $z=E+\rm i\eta_{0}$ with $\eta_{0}=n^{-2/3-\epsilon}$. By Cauchy’s
integral formula, we have

$\displaystyle\mathbf{y}_{\gamma}^{*}(G_{\gamma}^{(\gamma)})^{2}\mathbf{y}_{\gamma}=\frac{1}{2\pi\rm
i}\oint_{\Gamma_{\eta_{0}}}\frac{\mathbf{y}_{\gamma}^{*}G_{\gamma}^{(\gamma)}(x)\mathbf{y}_{\gamma}}{(x-z)^{2}}\mathrm{d}x,$

where $\Gamma_{\eta_{0}}$ is the disk centered at $z$ with radius
$\eta_{0}/2$. Decompose that

$\displaystyle|\mathbf{y}_{\gamma}^{*}(G_{\gamma}^{(\gamma)})^{2}\mathbf{y}_{\gamma}|\lesssim\left|\oint_{\Gamma_{\eta_{0}}}\frac{\xi^{2}_{\gamma}\operatorname{tr}G_{\gamma}^{(\gamma)}(x)}{p(x-z)^{2}}\mathrm{dx}\right|+\left|\oint_{\Gamma_{\eta_{0}}}\frac{\mathbf{y}_{\gamma}^{*}G_{\gamma}^{(\gamma)}(x)\mathbf{y}_{\gamma}-\xi^{2}_{\gamma}p^{-1}\operatorname{tr}G_{\gamma}^{(\gamma)}(x)}{(x-z)^{2}}\mathrm{d}x\right|.$

For the first term, we see from Cauchy’s integral formula, Theorem 3.4, Lemmas
A.2, A.3 and 3.2 that

$\displaystyle\left|\oint_{\Gamma_{\eta_{0}}}\frac{\xi^{2}_{\gamma}\operatorname{tr}G_{\gamma}^{(\gamma)}(x)}{p(\mathrm{x}-z)^{2}}\mathrm{dx}\Big{|}\lesssim\Big{|}\frac{\xi^{2}_{\gamma}\operatorname{tr}(G_{\gamma}^{(\gamma)}(z))^{2}}{p}\right|\lesssim\frac{1}{p}\sum_{i=1}^{p}\frac{1}{|\lambda_{i}^{(\gamma)}-z|^{2}}=\frac{\operatorname{Im}\operatorname{tr}\mathcal{G}_{\gamma}^{(\gamma)}}{p\eta_{0}}\prec\frac{1}{\sqrt{\eta_{0}}}=n^{1/3+\epsilon}.$

With similar reasoning, for the second term, we have that

$\displaystyle\left|\oint_{\Gamma_{\eta_{0}}}\frac{\mathbf{y}_{\gamma}^{*}G_{\gamma}^{(\gamma)}(x)\mathbf{y}_{\gamma}-\xi^{2}_{\gamma}p^{-1}\operatorname{tr}G_{\gamma}^{(\gamma)}(x)}{(x-z)^{2}}\mathrm{dx}\right|\prec\Big{(}\frac{\operatorname{Im}\operatorname{tr}G_{\gamma}^{(\gamma)}}{p^{2}\eta_{0}^{3}}\Big{)}^{1/2}\lesssim\Big{(}\frac{\operatorname{Im}\operatorname{tr}\mathcal{G}_{\gamma}^{(\gamma)}}{p^{2}\eta_{0}^{3}}\Big{)}^{1/2}\prec
n^{1/3+\epsilon}.$

This completes our proof. ∎

###### Proof of (4.9).

For the diagonal case, we only show the result for $(G_{0}(z))_{11}$ when
$z=E+\mathrm{i}\eta_{0}$ with $\eta_{0}=n^{-2/3-\epsilon}$,
$|E-\lambda_{+}|\leq n^{-2/3+\epsilon}$ for $\gamma=0$, while other cases
$(G_{\gamma}^{(\gamma)}(z))_{ii},\gamma\geqslant 1$ can be handled similarly.
By resolvent identity (see equation (5.5) of [21]), one has

$\displaystyle(G_{0}(z))_{11}=\frac{1}{-z(1+\mathbf{r}_{1}^{*}\mathcal{G}_{0}^{[1]}(z)\mathbf{r}_{1})}=\frac{1}{-z\big{(}1+\sigma_{1}(m_{2n}^{[1]}(z)+\mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3})\big{)}},$

where
$\mathbf{r}_{i}:=(\xi_{1}w_{i1}\sqrt{\sigma_{i}},\xi_{2}w_{i2}\sqrt{\sigma_{i}},\dots,\xi_{n}w_{in}\sqrt{\sigma_{i}})$
is $i$-th row of $V$ as in (2.14) and the upper index $[i]$ denotes the $i$-th
row of the corresponding data matrix $Y$ being deleted, and
$\mathrm{R}_{3}:={\sum_{i\neq
j}\xi_{i}\xi_{j}w_{1i}w_{1j}(\mathcal{G}_{0}^{[1]}(z))_{ij}}$ and

$\displaystyle\mathrm{R}_{1}:={-z^{-1}\sum_{i}\xi^{2}_{i}w_{1i}^{2}(1+m_{1n}^{[1]}(z)D^{2})^{-1}_{ii}}-m_{2n}^{[1]}(z),\quad\mathrm{R}_{2}:={\sum_{i}\xi^{2}_{i}w_{1i}^{2}\big{(}(\mathcal{G}_{0}^{[1]}(z))_{ii}}+z^{-1}(1+m_{1n}^{[1]}(z)D^{2})^{-1}_{ii}\big{)}.$

It is easy to see that Theorem 3.4 still applies to $\mathcal{G}_{0}^{[1]}(z)$
so that

$\displaystyle\Big{|}({\mathcal{G}_{0}^{[1]}(z)})_{ij}+z^{-1}(1+m_{1n}^{[1]}(z)D^{2})^{-1}_{ij}\Big{|}\prec\sqrt{\frac{\operatorname{Im}m_{1n}^{[1]}(z)}{p\eta}}+\frac{1}{p\eta}.$

For $\mathrm{R}_{3}$, let $\\{\varepsilon_{i}\\}$ be i.i.d. Rademacher random
variables, we see that $\mathrm{R}_{3}\overset{d}{=}\sum_{i\neq
j}\xi_{i}\xi_{j}\varepsilon_{i}\varepsilon_{j}{w_{1i}w_{1j}}({\mathcal{G}_{0}^{[1]}(z)})_{ij}.$
Therefore, by large deviation of the quadratic form of $\varepsilon_{i}$’s, we
have

$\displaystyle\left|\sum_{i\neq
j}\xi_{i}\xi_{j}\varepsilon_{i}\varepsilon_{j}{w_{1i}w_{1j}}({\mathcal{G}_{0}^{[1]}(z)})_{ij}\right|\prec\Big{(}\sum_{i\neq
j}\xi^{2}_{i}\xi^{2}_{j}{w^{2}_{1i}w^{2}_{1j}}\big{(}({\mathcal{G}_{0}^{[1]}(z)})_{ij}\big{)}^{2}\Big{)}^{1/2}\prec
p^{-1/3+\epsilon}.$ (4.17)

It follows that $\mathrm{R}_{3}\prec p^{-1/3+\epsilon}$. Next, for
$\mathrm{R}_{2}$, we have

$\displaystyle|\mathrm{R}_{2}|\leq\sum_{i}\xi^{2}_{i}{w_{1i}^{2}}\left|{(\mathcal{G}_{0}^{[1]}(z)})_{ii}+z^{-1}(1+m_{1n}^{[1]}(z)D^{2})^{-1}_{ii}\right|\prec
p^{-1/3+\epsilon}\sum_{i}\xi^{2}_{i}{w_{1i}^{2}}\prec n^{-1/3+\epsilon}.$

Lastly, for $\mathrm{R}_{1}$, using the fact $|m_{2n}^{[1]}(z)|\sim 1$, we
have the trivial bound $|\mathrm{R}_{1}|\prec n^{-1/2}$ since $w_{1i}^{2}$’s
are i.i.d.. Then by the definition of $m_{2n}^{[1]}$ and Lemma A.2, together
with above estimates, we have
$(G_{0}(z))_{11}=-z^{-1}(1+\sigma_{1}m_{2n}(z))^{-1}+\mathrm{O}_{\prec}(n^{-1/3+\epsilon}).$
For $i\neq j$, by the resolvent identity (see equation (5.6) of [21]), one has

$\displaystyle|(G_{0}(z))_{ij}|=|z(G_{0}(z))_{ii}(G^{[1]}_{0}(z))_{jj}\mathbf{r}_{i}^{*}\mathcal{G}^{[ij]}_{0}(z)\mathbf{r}_{j}|\prec|\mathbf{r}_{i}^{*}\mathcal{G}^{[ij]}(z)\mathbf{r}_{j}|=\left|\sum_{k,l}\xi_{k}\xi_{l}w_{ik}w_{jl}\sqrt{\sigma}_{k}\sqrt{\sigma}_{l}(\mathcal{G}_{0}^{[ij]}(z))_{kl}\right|.$

Then we can conclude the proof using a discussion similar to (4.17) with the
help of Theorem 3.4. ∎

###### Proof of (4.10).

As in the proof of (4.9), we only consider $\gamma=0.$ By Cauchy’s integral
formula, we have

$\displaystyle|((G_{0}(z))^{2})_{ij}|\lesssim\Big{|}\oint_{\Gamma_{\eta_{0}}}\frac{(G_{0}(x))_{ij}}{(x-z)^{2}}\mathrm{d}x\Big{|}.$

For the case $i\neq j$, we obtain straight forwards from (4.9) that

$\displaystyle\Big{|}\oint_{\Gamma_{\eta_{0}}}\frac{(G_{0}(x))_{ij}}{(x-z)^{2}}\mathrm{d}x\Big{|}\prec
n^{1/3+\epsilon}.$

If $i=j$, we see from Lemma 3.2 that $|m_{2n}^{\prime}(z)|\lesssim
n^{1/3+\epsilon}$. Then we have

$\displaystyle\begin{split}&\left|\oint_{\Gamma_{\eta_{0}}}\frac{{(G_{0}(x))_{ii}}}{(x-z)^{2}}\mathrm{d}x\right|\lesssim\left|\oint_{\Gamma_{\eta_{0}}}\frac{{(G_{0}(x))_{ii}+{x^{-1}(1+\sigma_{i}m_{2n}(x))^{-1}}}}{(x-z)^{2}}\mathrm{d}x\right|+\left|\oint_{\Gamma_{\eta_{0}}}\frac{{{x^{-1}(1+\sigma_{i}m_{2n}(x))^{-1}}}}{(x-z)^{2}}\mathrm{d}x\right|\\\
&\prec\frac{n^{-1/3+\epsilon}}{\eta_{0}}+\mathrm{O}(m_{2n}^{\prime}(z))\prec
n^{1/3+2\epsilon}.\end{split}$

This completes the proof of (4.10). ∎

##  Appendix A Proof of the local laws

We prove Theorem 3.4 following [21]. Due to similarity, we only provide the
key ingredients of the counterparts of [21].

###  A.1 Basic tools and auxiliary lemmas

In this section we collect some necessary notations and technical tools.
Denote the singular value decomposition of $V$ in (2.14) as
$V=\sum_{k=1}^{p\wedge n}\sqrt{\lambda_{k}}\gamma_{k}\zeta_{k}^{*}.$ Then for
the resolvents in (3.8), we can get for $1\leqslant i,j\leqslant p$ and
$1\leqslant\mu,\nu\leqslant n$,

$\displaystyle
G_{ij}=\sum_{k=1}^{p}\frac{z\gamma_{k}(i)\gamma^{*}_{k}(j)}{\lambda_{k}-z},\quad\mathcal{G}_{ji}=\sum_{k=1}^{n}\frac{\zeta_{k}(j)\zeta^{*}_{k}(i)}{\lambda_{k}-z}.$
(A.1)

First, we will frequently use the following identities whose proof can be
found in [4, 19, 21].

######  Lemma A.1.

For any $\mathcal{T}\subset\\{1,2,\cdots,n\\}$, we have that

$\displaystyle\mathcal{G}_{ii}^{(\mathcal{T})}(z)$ $\displaystyle=$
$\displaystyle-\frac{1}{z+z\mathbf{v}_{i}^{*}{G}^{(i\mathcal{T})}(z)\mathbf{v}_{i}},\qquad\forall
i\in\mathcal{I}\setminus\mathcal{T},$
$\displaystyle\mathcal{G}_{ij}^{(\mathcal{T})}(z)$ $\displaystyle=$
$\displaystyle
z\mathcal{G}_{ii}^{(\mathcal{T})}(z)\mathcal{G}_{jj}^{(i\mathcal{T})}(z)\mathbf{v}_{i}^{*}{G}^{(ij\mathcal{T})}(z)\mathbf{v}_{j},\qquad\forall
i,j\in\mathcal{I}\setminus\mathcal{T},i\neq j,$
$\displaystyle\mathcal{G}_{ij}^{(\mathcal{T})}(z)$ $\displaystyle=$
$\displaystyle\mathcal{G}_{ij}^{(k\mathcal{T})}(z)+\frac{\mathcal{G}_{ik}^{(\mathcal{T})}(z)\mathcal{G}_{kj}^{(\mathcal{T})}(z)}{\mathcal{G}_{kk}^{(\mathcal{T})}(z)},\qquad\forall
i,j,k\in\mathcal{I}\setminus\mathcal{T},i,j\neq k.$

Moreover,

$\displaystyle\sum_{1\leq i\leq p}|G_{ji}|^{2}=\sum_{1\leq i\leq
p}|G_{ij}|^{2}=\frac{|z|^{2}}{\eta}\operatorname{Im}(\frac{G_{jj}}{z}),\
\sum_{1\leq\mu\leq n}|\mathcal{G}_{\nu\mu}|^{2}=\sum_{1\leq\mu\leq
n}|\mathcal{G}_{\mu\nu}|^{2}=\frac{\operatorname{Im}\mathcal{G}_{\nu\nu}}{\eta},$
$\displaystyle\|G\Sigma\|_{F}^{2}=\eta^{-1}\operatorname{Im}\operatorname{tr}(G\Sigma^{2}).$

Second, the following estimates are also for our discussions whose proof can
also be found in [4, 19, 21].

######  Lemma A.2.

The following estimates hold uniformly for all $z\in\mathbf{D}$ in (3.2) and
$C>0$,

$\displaystyle\|G\|+\|\mathcal{G}\|\leq
C\eta^{-1},\quad\|\partial_{z}G\|+\|\partial_{z}\mathcal{G}\|\leq C\eta^{-2}.$

The above estimates remain true for $G^{(\mathcal{T})}$ instead of $G$ for any
$\mathcal{T}\subset\\{1,\dots,p\\}$

$\displaystyle|m_{1}-m_{1}^{(\mathcal{T})}|+|m_{2}-m_{2}^{(\mathcal{T})}|\leq\frac{C|\mathcal{T}|}{n\eta}.$

Finally, the following estimates will be used in our calculations whose proof
can be found in [4, 20].

######  Lemma A.3.

(1). Let $\mathbf{u}=(u_{1},\dots,u_{p})$ be a $p$-dimensional random vector
of spherical uniform distribution. For any $i\neq j\in\\{1,\dots,p\\}$, let
$k_{1},k_{2}$ be two non-negative integers, we have

$\displaystyle\mathbb{E}|u_{i}^{2}|=\frac{1}{p},\quad\mathbb{E}|u_{i}^{4}|=\frac{3}{p(2+p)},\quad\mathbb{E}|u_{i}^{2}u_{j}^{2}|=\frac{1}{p(2+p)},\quad\mathbb{E}|u_{i}^{k_{1}}u_{j}^{k_{2}}|=0,\quad\text{if
$k_{1}+k_{2}$ is odd.}$ (A.2)

(2). $\mathbf{w}=(w_{1},\dots,w_{p})^{*}$,
$\widetilde{\mathbf{w}}=(\widetilde{w}_{1},\dots,\widetilde{w}_{p})^{*}$ be
two independent random vectors as in (2.14). Suppose $A=(a_{ij})$ an $p\times
p$ matrix and $\mathbf{b}=(b_{1},\dots,b_{p})^{*}$ an $p$-dimensional vector,
where $A$ and $\mathbf{b}$ may be complex-valued and
$\mathbf{w},\tilde{\mathbf{w}},A,\mathbf{b}$ are independent. Then as
$p\to\infty$

$\displaystyle|\mathbf{b}^{*}\mathbf{w}|\prec\sqrt{\frac{\|\mathbf{b}\|^{2}}{p},}\
|\mathbf{w}^{*}A\mathbf{w}-\frac{1}{p}{\rm tr}A|\prec\frac{1}{p}\|A\|_{F},\
\Big{|}\mathbf{w}^{*}A\widetilde{\mathbf{w}}\Big{|}\prec\frac{1}{p}\|A\|_{F}.$

###  A.2 Proof of Theorem 3.4

For simplicity, following [3, 5, 21] we introduce the following notations.

$\displaystyle\Pi_{1}(z):=-(1+m_{2n}(z)\Sigma)^{-1},\quad\Pi_{2}(z):=-z^{-1}(1+m_{1n}(z)D^{2})^{-1}.$

Note that $\frac{1}{zp}\operatorname{tr}\Pi_{1}=m_{n}(z).$ Moreover, we denote
$m_{2}(z):=p^{-1}\sum_{i=1}^{n}\xi^{2}_{i}\mathcal{G}_{ii}(z).$ In addition,
it is more convenient to introduce the following parameters

$\displaystyle\Lambda\equiv\Lambda(z):=\max_{i,j}|\mathcal{G}-\Pi_{2}|_{ij},\quad\Theta\equiv\Theta(z):=|m_{1}(z)-m_{1n}(z)|+|m_{2}(z)-m_{2n}(z)|,$
$\displaystyle\Lambda_{o}\equiv\Lambda_{o}(z):=\max_{i\neq
j}|\mathcal{G}_{ij}|,\quad\Psi_{\Theta}\equiv\Psi_{\Theta}(z):=\sqrt{\frac{\operatorname{Im}m_{1n}(z)+\Theta}{p\eta}}+\frac{1}{p\eta}.$

It is easy to see that on the event $\Xi:=\\{\Lambda\leq(\log p)^{-1}\\},$ the
following holds uniformly on $i,j\in\\{1,2,\cdots,n\\}$ and $z\in\mathbf{D}$

$\displaystyle\\{\mathbf{1}(\Xi)+\mathbf{1}(\eta\geq
1)\\}|\mathcal{G}_{ij}^{(\mathcal{T)}}|+\mathbf{1}(\Xi)|(\mathcal{G}^{(\mathcal{T})}_{ii})^{-1}|=\mathrm{O}_{\prec}(1).$
(A.3)

Moreover, by (3.6) and (3.7), when $\Theta\prec(n\eta)^{-1}$,

$\displaystyle\|\Pi_{1,2}\|=\mathrm{O}(1),\quad\Psi_{\Theta}\gtrsim
p^{-1/2},\quad\Psi_{\Theta}^{2}\lesssim(p\eta)^{-1},\quad\Psi_{\Theta}(z)\sim\sqrt{\frac{\operatorname{Im}m_{1n}(z)}{p\eta}}+\frac{1}{p\eta},\quad
z\in\mathbf{D}_{0}.$

Furthermore, we introduce the $Z$ variable

$\displaystyle
Z_{i}^{(\mathcal{T})}:=(1-\mathbb{E}_{i})\big{[}\mathbf{v}_{i}^{*}G^{(i\mathcal{T})}\mathbf{v}_{i}\big{]},\quad
i\notin\mathcal{T},$

where $\mathbb{E}_{i}[\cdot]:=\mathbb{E}[\cdot|W^{(i)}]$ is the partial
expectation over the randomness of the $i$-th row and column of $W$ in (2.14).
Thanks to (A.2), we see that observe that for both choice of $\mathbf{w}_{i}$,
it follows

$\displaystyle
Z_{i}=\mathbf{v}_{i}^{*}G^{(i)}\mathbf{v}_{i}-\frac{\xi^{2}_{i}}{p}\operatorname{tr}(G^{(i)}\Sigma).$

The following lemma is a counterpart of Lemma 5.9 of [21].

######  Lemma A.4.

Suppose assumptions of Theorem 3.4 hold. Then for all $1\leqslant i\leqslant
n$ and $z\in\mathbf{D}$ uniformly

$\displaystyle\\{\mathbf{1}(\Xi)+\mathbf{1}(\eta\geq
1)\\}(|Z_{i}|+\Lambda_{o})\prec\Psi_{\Theta}.$

###### Proof.

Let $\mathcal{I}=\\{1,2,3,\cdots,n\\}.$ Applying Lemmas A.1, A.3 and 3.2, we
obtain that uniformly for $z\in\mathbf{D}$ and $i,j\in\mathcal{I}$ with $i\neq
j$,

$\displaystyle\mathbf{1}(\Xi)|\mathcal{G}_{ij}|\leq\mathbf{1}(\Xi)|z||\xi_{i}\xi_{j}||\mathcal{G}_{ii}\mathcal{G}_{jj}^{(i)}||\mathbf{v}_{i}^{*}G^{(ij)}\mathbf{v}_{j}|\prec\mathbf{1}(\Xi)|\mathcal{G}_{ii}\mathcal{G}_{jj}^{(i)}||\xi_{i}\xi_{j}|\frac{1}{p}\|\Sigma^{1/2}\|\|G^{(ij)}\Sigma^{1/2}\|_{F}.$
(A.4)

Together with Lemmas A.1 and A.2, we can see that

$\displaystyle\mathbf{1}(\Xi)|\mathcal{G}_{ij}|\prec\sqrt{\frac{\operatorname{Im}m_{1}^{(ij)}}{p\eta}}\prec\sqrt{\frac{\operatorname{Im}m_{1n}+\Theta}{p\eta}}+\frac{1}{p\eta}.$

Therefore, $\mathbf{1}(\Xi)\Lambda_{o}\prec\Psi_{\Theta}$. The result for
$Z_{i}$ under $\Xi$ can be derived similarly in the sense that

$\displaystyle\mathbf{1}(\Xi)Z_{i}=\mathbf{1}(\Xi)\big{(}\mathbf{v}_{i}^{*}G^{(i)}\mathbf{v}_{i}-\frac{\xi^{2}_{i}}{p}\operatorname{tr}(G^{(i)}\Sigma)\big{)}\prec\mathbf{1}(\Xi)\frac{\xi^{2}_{i}}{p}\|\Sigma^{1/2}\|\|G^{(i)}\Sigma^{1/2}\|_{F}\prec\Psi_{\Theta}.$

Similar procedure can be applied to $\mathcal{G}_{ij}$ and $Z_{i}$ via (A.3).
This completes the proof. ∎

The following lemma is a counterpart of Lemma 5.10 of [21].

######  Lemma A.5.

Suppose assumptions in Theorem 3.4 hold. Then

$\displaystyle\mathbf{1}(\eta\geq 1)|F_{p}(m_{1}(z),z)|\prec
p^{-1/2},\quad\mathbf{1}(\eta\geq
1)\left|m_{2}(z)+\frac{1}{p}\sum_{i=1}^{n}\frac{\xi^{2}_{i}}{z(1+\xi^{2}_{i}m_{1}(z))}\right|\prec
p^{-1/2},$ (A.5)
$\displaystyle\mathbf{1}(\Xi)|F_{p}(m_{1}(z),z)|\prec\Psi_{\Theta},\quad\mathbf{1}(\Xi)\left|m_{2}(z)+\frac{1}{p}\sum_{i=1}^{n}\frac{\xi^{2}_{i}}{z(1+\xi^{2}_{i}m_{1}(z))}\right|\prec\Psi_{\Theta}.$
(A.6)

Moreover, we have the finer estimates

$\displaystyle\mathbf{1}(\Xi)|F_{p}(m_{1}(z),z)|\prec\mathbf{1}(\Xi)\Big{(}|\mathbf{Z}_{1}|+|\mathbf{Z}_{2}|\Big{)}+\Psi_{\Theta}^{2},$
(A.7)

and

$\displaystyle\mathbf{1}(\Xi)|m_{2}(z)+\frac{1}{p}\sum_{i=1}^{n}\frac{\xi^{2}_{i}}{z(1+\xi^{2}_{i}m_{1}(z))}|\prec\mathbf{1}(\Xi)\big{(}|\mathbf{Z}_{2}|\big{)}+\Psi_{\Theta}^{2},$
(A.8)

where

$\displaystyle\mathbf{Z}_{1}:=\frac{1}{p}\sum_{j=1}^{n}\frac{\xi^{2}_{j}}{z(1+\xi^{2}_{j}m^{(j)}_{1}(z))}Z_{j},\quad\mathbf{Z}_{2}:=\frac{1}{p}\sum_{j=1}^{n}\frac{\xi^{2}_{j}}{z(1+\xi^{2}_{j}m^{(j)}_{1}(z))^{2}}Z_{j}.$

###### Proof.

We begin the proof with the finer estimates (A.7) and (A.8). By Lemmas A.1,
A.2, A.3 and 3.2

$\displaystyle\mathbf{1}(\Xi)\mathcal{G}_{jj}=\mathbf{1}(\Xi)\Big{(}-\frac{1}{z(1+\xi_{j}^{2}m_{1}(z))}+\frac{Z_{j}}{z(1+\xi^{2}_{j}m_{1}^{(j)}(z))^{2}}+\mathrm{O}_{\prec}(\Psi_{\Theta}^{2})\Big{)},$
(A.9)

which results in (A.8) by taking average with $p^{-1}\sum_{j}\xi^{2}_{j}$.
Moreover, using (B.14) of [3], we see that

$\displaystyle\begin{split}G&=-z^{-1}(I+m_{2}(z)\Sigma)^{-1}+z^{-1}\sum_{i=1}^{n}\frac{G^{(i)}(\mathbf{v}_{i}\mathbf{v}_{i}^{*}-p^{-1}\xi^{2}_{i}\Sigma)}{1+\mathbf{v}_{i}^{*}G^{(i)}\mathbf{v}_{i}}(I+m_{2}(z)\Sigma)^{-1}+z^{-1}\frac{1}{p}\sum_{i=1}^{n}\frac{(G^{(i)}-G)\xi^{2}_{i}\Sigma}{1+\mathbf{v}_{i}^{*}G^{(i)}\mathbf{v}_{i}}(I+m_{2}(z)\Sigma)^{-1}\\\
&:=-z^{-1}(I+m_{2}(z)\Sigma)^{-1}+R_{1}+R_{2}.\end{split}$

Taking average with $p^{-1}\sum_{i}\sigma_{i}$, we have

$\displaystyle
m_{1}(z)=\frac{1}{p}\operatorname{tr}(G(z)\Sigma)=-\frac{1}{p}\sum_{i=1}^{p}\frac{\sigma_{i}}{z(1+m_{2}(z)\sigma_{i})}+\frac{1}{p}\operatorname{tr}(R_{1}\Sigma)+\frac{1}{p}\operatorname{tr}(R_{2}\Sigma).$
(A.10)

Similar to (A.9), we have that

$\displaystyle\mathbf{1}(\Xi)\Big{(}\frac{1}{p}\operatorname{tr}(R_{1}\Sigma)\Big{)}=\mathbf{1}(\Xi)\mathrm{O}_{\prec}(\mathbf{Z}_{1}+\Psi^{2}_{\Theta}),\quad\mathbf{1}(\Xi)\Big{(}\frac{1}{p}\operatorname{tr}(R_{2}\Sigma)\Big{)}=\mathbf{1}(\Xi)\mathrm{O}_{\prec}(\mathbf{Z}_{2}+\Psi^{2}_{\Theta}).$

This results in (A.7) using (2.8).

Armed with the above results, (A.6) follows from Lemma A.4. For (A.5), the
arguments are similar except we need to prove that in the case $\eta\geq 1$,
with high probability, for all $1\leqslant i\leqslant p,1\leqslant j\leqslant
n$ and some constant $c^{\prime}>0$

$\displaystyle|1+\xi^{2}_{j}m_{1}|\geq c^{\prime},\quad|1+\sigma_{i}m_{2}|\geq
c^{\prime}.$ (A.11)

The proof of (A.11) follows lines between equations (5.43) and (5.45) of [21]
and we omit the details. This concludes our proof. ∎

The following lemma is a counterpart of Lemma 5.11 of [21].

######  Lemma A.6.

Suppose assumptions in Theorem 3.4 hold. Suppose a $z$-dependent function
$\delta$ satisfying $p^{-1}\leq\delta(z)\leq\log^{-1}p$ for $z\in\mathbf{D}$
and assume that $\delta(z)$ is Lipschitz continuous with Lipschitz constant
$\leq p^{2}$. Suppose moreover that for each fixed $E$, the function
$z\mapsto\delta(E+\mathrm{i}\eta)$ is non-increasing for $\eta>0$. Suppose
that $\mu_{0}:\mathbf{D}\rightarrow\mathbb{C}$ is the Stieltjes transform of a
probability measure. Let $z\in\mathbf{D}$ and suppose that for all
$z^{\prime}\in\operatorname{Lip}(z)$, we have
$|F_{p}(\mu_{0},z)|\leq\delta(z^{\prime}),$ then we have that for some
constant $C>0$

$\displaystyle|\mu_{0}-m_{1n}(z)|\leq\frac{C\delta}{\sqrt{\kappa+\eta+\delta}}.$

###### Proof.

The proof of this lemma of relies on the first and second statements in Lemma
3.2. One can refer to [21, Lemma 5.11] for more details. ∎

With Lemma A.6, we see from (A.5), (A.6) and (A.11) that

$\displaystyle\mathbf{1}(\eta\geq 1)\Theta(z)\prec p^{-1/2}.$ (A.12)

Then from Lemma A.4, we have for the off-diagonal entries,
$\mathbf{1}(\eta\geq 1)\Lambda_{o}\prec p^{-1/2}.$ For the diagonal entries,
using Lemma A.4, (A.12) and (A.11) with the following expression

$\displaystyle\mathbf{1}(\eta\geq 1)\mathcal{G}_{ii}=\mathbf{1}(\eta\geq
1)\Big{(}-\frac{1}{z(1+\xi^{2}_{i}m_{1})}+\frac{Z_{i}}{z(1+\xi^{2}_{i}m_{1})^{2}}+\mathrm{O}_{\prec}(\Psi_{\Theta}^{2})\Big{)},$

we obtain that $\mathbf{1}(\eta\geq 1)(|\mathcal{G}-\Pi_{2}|_{ii})\prec
p^{-1/2}$ for all $1\leqslant i\leqslant n.$ This yields that
$\mathbf{1}(\eta\geq 1)\Lambda(z)\prec p^{-1/2}.$ It remains to deal with the
small $\eta<1$. The following weak bound is a counterpart of Lemma 5.12 of
[21].

######  Lemma A.7 (Weak entrywise local law).

Suppose assumptions in Theorem 3.4 hold. Then we have
$\Lambda(z)\prec(p\eta)^{-1/4}$ uniformly for $z\in\mathbf{D}$.

###### Proof.

One can prove this lemma using a continuity argument. The key inputs are the
estimates in the $\eta\geqslant 1$ case. All the other parts of the proof are
essentially the same as Lemma 5.12 of [21]. We omit the details. ∎

The following lemma is the last component for the proof of (3.9) which is a
counterpart of Lemma 5.13 of [21].

######  Lemma A.8 (Fluctuation Averaging).

Suppose assumptions in Theorem 3.4 hold. Let $\nu\in[1/4,1]$. Denote
$\Psi_{\nu}:=\sqrt{\frac{\operatorname{Im}m_{1n}+(p\eta)^{-\nu}}{p\eta}}+\frac{1}{p\eta}$.
Suppose moreover that $\Lambda\prec(p\eta)^{-\nu}$ uniformly for
$z\in\mathbf{D}$. Then we have that
$|\mathbf{Z}_{1}|+|\mathbf{Z}_{2}|\prec\Psi_{\nu}^{2}.$

###### Proof.

Notice that $\Xi$ holds with high probability from Lemma A.7. We can write

$\displaystyle\mathbf{Z}_{1}=\frac{1}{p}\sum_{j=1}^{n}(1-\mathbb{E}_{j})\Big{[}\frac{\xi^{2}_{j}}{1+\xi^{2}_{j}m_{1}^{(j)}(z)}\mathbf{v}_{j}^{*}G^{(j)}\mathbf{v}_{j}\Big{]},\quad\mathbf{Z}_{2}=\frac{1}{p}\sum_{j=1}^{n}(1-\mathbb{E}_{j})\Big{[}\frac{\xi^{2}_{j}}{(1+\xi^{2}_{j}m_{1}^{(j)}(z))^{2}}\mathbf{v}_{j}^{*}G^{(j)}\mathbf{v}_{j}\Big{]}.$

Now the method to prove the fluctuation averaging for the random terms under
conditional expectation is relatively standard; for example see Lemma 5.13 of
[21] or Proposition 4.11 of [20]. We omit the further details here. ∎

With the above preparation, we prove Theorem 3.4. We start with the entry-wise
local law as in (3.9) following that of Proposition 5.8 of [21].

###### Proof of (3.9).

By Lemma A.7, the event $\Xi$ holds with high probability. According to Lemmas
A.4 and A.7, we may take $\nu=1/4$. Then, (A.7) reads that
$\mathbf{1}(\Xi)|F_{p}(m_{1}(z),z)|\prec(\operatorname{Im}m_{1n}(z)+(p\eta)^{-1/4})(p\eta)^{-1}.$
Using the results in Theorem A.6, one has

$\displaystyle|m_{1}-m_{1n}|\prec\frac{\operatorname{Im}m_{1n}}{p\eta\sqrt{\kappa+\eta}}+\frac{1}{(p\eta)^{5/8}}\prec\frac{1}{(p\eta)^{5/8}},$

where we used $\operatorname{Im}m_{1n}(z)=\mathrm{O}(\sqrt{\kappa+\eta})$ from
(3.6). Similar bound can be derived for $|m_{2}-m_{2n}|$ by (A.8). Then we get
an updated bound $\Theta\prec(p\eta)^{-5/8}.$ Plugging such updated bound in
$\Psi_{\Theta}$ in Lemma A.4, one has

$\displaystyle\Lambda_{o}\prec\sqrt{\frac{\operatorname{Im}m_{1n}(z)+(p\eta)^{-5/8}}{p\eta}}+\frac{1}{p\eta},$

uniformly in $z\in\mathbf{D}$. Up to now, we obtain a better bound for
$\Lambda_{o}$. Iterating the above arguments as in the discussions between
(5.57) and (5.58) of [21], we get the bound

$\displaystyle\Theta\prec(p\eta)^{-1}.$ (A.13)

Combining (A.9) with (A.13) and Lemma A.8, we can finish the proof. ∎

Then we prove the averaged local laws as in (3.10) and (3.11) following that
of Proposition 5.1 of [21].

###### Proof of (3.10) and (3.11).

By a discussion similar to (A.10), we can obtain that

$\displaystyle
m(z)=-\frac{1}{p}\sum_{i=1}^{p}\frac{1}{z(1+m_{2}(z)\sigma_{i})}+\mathrm{O}_{\prec}(\mathbf{Z}_{2}+\Psi_{\Theta}^{2}).$
(A.14)

Using (2.5), (3.7), Lemma A.8 and (A.13), we can prove (3.10). Moreover, for
$z\in\mathbf{D}\cap\\{z=E+\mathrm{i}\eta:E\geq\lambda_{+},\;p\eta\sqrt{\kappa+\eta}\geq
p^{\varepsilon_{e}}\\}$, using (3.6), we observe that

$\displaystyle\Psi_{\Theta}^{2}\leq
2\Big{(}\frac{\operatorname{Im}m_{1n}(z)}{p\eta}+\frac{1}{(p\eta)^{2}}\Big{)}\lesssim\frac{1}{p\sqrt{\kappa+\eta}}+\frac{1}{(p\eta)^{2}}.$
(A.15)

Taking $\nu=1$ in Lemma A.8, we can prove (3.11) using Lemma A.6. This
completes the proof. ∎

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