ZhidongBai JiangHu GuangmingPan WangZhou ∗ ANoteonRateofConvergenceinProbabilitytoSemicircularLaw

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 16 (2011), Paper no. 88, pages 2439–2451.

Journal URL

http://www.math.washington.edu/~ejpecp/

A Note on Rate of Convergence in Probability to Semicircular Law

Zhidong Bai^† Jiang Hu^‡ Guangming Pan^§ Wang Zhou^¶

Abstract

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability isO(n^−1/2)when the dimensionntends to infinity..

Key words:convergence rate, Wigner matrix, Semicircular Law, spectral distribution.

AMS 2010 Subject Classification:Primary 60F15; Secondary: 62H99.

Submitted to EJP on March 18, 2010, final version accepted November 17, 2011.

∗Z. D. Bai was partially supported by CNSF 10871036. J. Hu was partially supported by the Fundamental Research Funds for the Central Universities 10ssxt149. G.M. Pan was partially supported by a grant M58110052 at the Nanyang Technological University. W. Zhou was partially supported by grant R-155-000-095-112 at the National University of Singapore

†KLASMOE and School of Mathematics & Statistics, Northeast Normal University, Changchun, 130024, P.R.C. and Department of Statistics and Applied Probability, National University of Singapore. E-mail: [email protected]

‡KLASMOE and School of Mathematics & Statistics, Northeast Normal University, Changchun, 130024, P.R.C. E-mail:

[email protected]

§Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University.

E-mail: [email protected]

¶Department of Statistics and Applied Probability, National University of Singapore, Singapore 117546. E-mail:

[email protected]

1 Introduction and the result.

A Wigner matrixW_n = n^−1/2€ x_{i j}Šn

i,j=1 is defined to be a Hermitian random matrix whose entries on and above the diagonal are independent zero-mean random variables. It is an important model for depicting heavy-nuclei atoms, which began with the seminal work of Wigner in 1955 ([16]).

Details in this area can be found in[13].

There are various mathematical tools in the study of Wigner matrices in the past half century (see [1]). One of the most popular instruments is the limit theory of empirical spectral distribution (ESD). Here, for anyn×nmatrixAwith real eigenvalues, the ESD ofAis defined by

F^A(x) = 1 n

n

X

i=1

I(λ^A_i ≤ x),

where λ^A_i denotes the i-th smallest eigenvalue of A and I(B)denotes the indicator function of an eventB. It is proved that, under assumptions of for alli,j,E|x_{i j}|²=σ², the ESDF^Wn(x)converges almost surely to a non-random distributionF(x)which has the destiny function

f(x) = 1 2πσ

p

4σ²−x², x ∈[−2σ, 2σ]. (1) This is also known as the Wigner semicircular law (see[16],[6]).

The rate of convergence is important in establishing the central limit theorem for linear spectral statistics of Wigner matrices ([7, 6]). There are some partial results in this area. In[2], Bai proved that under the assumption of sup_nsup_i,jE^x4_{i j}<∞, the rate of

∆n=kE^FWn−Fk:=sup

x |E^FWn(x)−F(x)|

tending to 0 isO(n^−1/4). Bai et al. in[4]obtained that the rate established in[2]was still valid in probability for

∆p=kF^Wn−Fk:=sup

x |F^Wn(x)−F(x)|

Under a stronger condition that sup_nsup_i,jE^x8_{i j} <∞, Bai et al. in[5]showed that∆n=O(n^−1/2) and∆_p=O_p(n^−2/5)(Bai and Silverstein improved this condition up to sup_nsup_i,jE^x_{i j}⁶ <∞in their book[6] ). Here and in the sequel, the notation R_n = O_p(r_n) means for any " > 0, there exists anC > 0 such that sup_nP(|R_n| ≥C r_n)< ". Later, Götze et al. in[10]derived∆_n =O(n^−1/2) as well assuming fourth moment, and∆p=O_p(n^−1/2)at the cost of the twelfth moment of the matrix entries. There are some other results with some special assumptions on the matrix entries. If the entries ofW_n have a normal distribution, then the optimal order∆_n=O(n⁻¹)was shown in[11]. When the distribution of the entries satisfies a Poincare inequality or a uniformly subexponential decay, the order of∆p can be improved toO_p(n^−2/3log²n)andO_p(n⁻¹log^Cn)with some constant C respectively. For which one can refer to[9, 8].

In this note we prove that the twelfth moment condition in [10] could be reduced to the sixth moment assumption while still getting∆p=O_p(n^−1/2). Our main result of this paper is as follows.

Theorem 1.1. Assume that

• Ex_{i j}=0, for all1≤i≤ j≤n,

• E|x²_ii|=σ²>0,E|x_{i j}|²=1, for all1≤i< j≤n,

• sup_nsup_1≤i<j≤nE|x³_ii|,E|x_{i j}|⁶<∞. Then we have

∆_p:=kF^Wn−Fk=O_p(n^−1/2). (2) Remark 1.2. It is not clear what the exact rate and the optimal conditions are in Theorem 1.1.

The rest of this paper is organized as follows. The main tool of proving the theorem is introduced in Section 2. Theorem 1.1 is proved in Section 3 and some technical lemmas are given in Sec- tion 4. Throughout this paper, constants appearing in inequalities are represented byC which are nonrandom and may take different values from one appearance to another.

2 The main tool

For any function of bounded variationH on the real line, its Stieltjes transform is defined by s_H(z) =

Z 1

λ−zd H(λ), z∈C⁺≡ {z∈C^+:ℑz>0}.

Our main tool to prove the theorem is a Berry-Esseen type inequality which is proved in[2]. Lemma 2.1. (Bai inequality) Let F be a distribution function and let G be a function of bounded variation satisfying R

|F(x)−G(x)|d x < ∞. Denote their Stieltjes transforms by s_F(z) and s_G(z) respectively, where z=u+i v∈C^{+. Then}

kF−Gk ≤ 1

π(1−ζ)(2ρ−1) Z A

−A

|s_F(z)−s_G(z)|du

+2πv⁻¹ Z

|x|>B

|F(x)−G(x)|d x

+v⁻¹sup

x

Z

|u|≤2vε

|G(x+u)−G(x)|du

! ,

where the constants A > B > 0, ζ and ε are restricted by ρ = _π¹R

|u|≤ε 1

u²+1du > ¹₂, and ζ =

4B

π(A−B)(2ρ−1)∈(0, 1).

Here we should notice that we can use the same methods in[10]to prove our theorem. However, Götze-Tikhomirov inequality (see Corollary 2.3 in[10]) involves the supremum of|s_n(z)−E^sn(z)|

overℑz in some interval. This makes the proof rather complicated. Therefore in this paper, we use Bai inequality instead of Götze-Tikhomirov inequality which could make the presentation simpler.

3 The proof of Theorem 1.1.

We will firstly introduce a new technique which can handle the moment conditions efficiently. That is given in Lemma 3.2. Then, by using this lemma and dividing the expression ofE|s_n−E^sn|², we prove our theorem step by step.

Before proving the theorem, we introduce some notation. DenoteI_n be the identity matrix of size nanda_i be theith column ofW_n with x_ii removed. DefineD(z) =n^−1/2W_n−zI_n,D_i(z) =D(z)− n⁻¹a_ia^∗_i ands_n=s_n(z) =s_F^Wn(z). Moreover write

βi=€

n^−1/2x_ii−z−n⁻¹a^∗_iD⁻_i ¹a_iŠ₋₁

, γi =a^∗_iD⁻_i ¹a_i−t rD⁻_i ¹

"_i=n^−1/2x_ii−n⁻¹a^∗_iD⁻_i ¹a_i+E^sn(z), ˆγ_i=a^∗_iD⁻_i²a_i−t rD⁻_i ² ξi =t rD⁻¹−t rD⁻¹_i , a_n= (z+E^sn(z))⁻¹.

Throughout this section, we denotez =u+i v, u∈[−16, 16]and 1≥ v ≥ v₀ = C₀n^−1/2 with an appropriate constantC₀. Lets=s(z) =s_F(z), we know that (see (3.2) in[2])

s(z) =−1 2

 z−

p z²−4

‹

for allz∈C^+. Then we have

Z 16

−16

1

|z+2s(z)|du≤ Z 16

−16

1 p|z²−4|

du≤ Z 16

−16

1 p|u²−4|

du<10. (3)

In addition, by Lemma 2.1 and Theorem 8.2 in[6], we have for some positive constantC, EkF^Wn−Fk ≤C

Z 16

−16

E|s_n(z)−E^sn(z)|du+O(n^−1/2). (4) Therefore, the rest of the proof is reduced to the lemma below.

Lemma 3.1. Under the assumptions in Theorem 1.1, for any1> v≥ v₀ =C₀n^−1/2 with sufficiently large C₀>0, we have

E^sn(z)−E^sn(z)

2≤ C

n|z+2s(z)|². 3.1 Known results and a preliminary lemma

Following the same truncation, centralization and rescaling steps in [6], in this section we may assume the random variables satisfy the conditions as follows

|x_{i j}| ≤n^1/4, E^xi j=0, E|x_{i j}|²=1 for alli,j.

Bai in[2]derived the equation

s_n(z) = 1

nt rD⁻¹= 1 n

Xn

i=1

βi,

which together with the fact

βi=−a_n+a_nβi"i, (5)

implies

s_n(z) =−a_n+a_n n

Xn i=1

βi"i. (6)

For eachiwe have

|ℑβ_i⁻¹|=|ℑ€

z+n⁻¹a^∗_iD⁻_i ¹a_iŠ

| ≥v.

Thus we have

|βi| ≤v⁻¹. (7)

From the definition of"i it follows that

"i =n^−1/2x_ii−n⁻¹γi+n⁻¹ξi−(sn−E^sn), (8) and

s_n=−a_n+ a_n n^3/2

Xn i=1

β_ix_ii+a_n n²

Xn i=1

β_iγ_i+a_n n²

Xn i=1

β_iξ_i−a_n(s_n−E^sn)s_n. (9)

Then, we have the the following lemma.

Lemma 3.2. Under the assumption in Theorem 1.1, we have for any v>v₀ P |βi|>2

≤ C

n²v². (10)

Proof. From integration by parts and Theorem 1.1 in[10], we have for 1>v>v₀,

|E^sn(z)−s(z)|=

Z _∞

−∞

d(E^FWn(x)−F(x)) x−z

=

Z _∞

−∞

E^FWn(x)−F(x) (x−z)² d x

≤C, which together with the fact that|s(z)| ≤1 ( see (3.3) in[2]) implies

|E^sn(z)| ≤C.

Then applying Lemma 4.2 and Lemma 4.3 we have for 1>v>v₀, E|s_n(z)| ≤C and 1

nE|t rD⁻_i¹| ≤C.

By Lemma 4.1 we can check that

E|γi|⁴≤CE^{€t rD−}_i ¹(D⁻_i ¹)^∗Š2

+n^1/2t r€

D⁻_i ¹(D⁻_i ¹)^∗Š2

≤C€

v⁻²E|t rD⁻_i ¹|²+n^1/2v⁻³E|t rD⁻_i¹|Š

≤ C n²

v² . (11)

Thus, from (8), Lemma 4.2 and Lemma 4.3 we have forv>v₀, E|"_i|⁴≤ C

n²v². (12)

In addition, by the proof of Lemma 5.1 and (5.59) in[10], we can see that

|a_n| ≤1 for allv≥v₀. (13)

Therefore from the equation (5) we can obtain P |βi|>2

≤P

|a_n"i|> 1 2

≤2⁴E|"i|⁴≤ C n²v², which complete the proof.

3.2 The proof of Lemma 3.1

Notice that in this subsection, we will use the equality (5) and (8) frequently. From (9), we have E^sn−E^sn

2=E(sn−E^sn)(sn−E^sn)

=E(sn−E^sn)sn=a_n(S1+S₂+S₃+S₄), where

S₁= 1 n^3/2

Xn

i=1

E(sn−E^sn)x_iiβi, S₂=− 1 n²

Xn

i=1

E(sn−E^sn)γiβi,

S₃= 1 n²

n

X

i=1

E(s_n−E^sn)ξiβi, S₄=−E|s_n−E^sn|²s_n.

We first considerS₁. From (5), we have S₁= 1

n^3/2

n

X

i=1

E(sn−Es_n)xiiβi

= a_n n^3/2

n

X

i=1

E^€−(s_n−E^sn)x_ii+a_nE(s_n−E^sn)x_ii"i−a_n(s_n−E^sn)x_iiβi"_i²Š

= a_n n^3/2

Xn

i=1

E −S₁₁+S₁₂−S₁₃ .

By Lemma 4.2 we have

|E^S11|= 1 nEξix_ii

≤E|x_ii| nv =O

1 n v

. (14)

From (12), (13), Hölder’s inequality and Lemma 4.3, we obtain

|E^S12| ≤€

E|s_n−E^sn|⁴E|"i|⁴(E|x_ii|²)²Š1/4

≤ C

n^3/2v². (15)

Next we consider the termS₁₃. Using (12), (13), Lemma 3.2, Lemma 4.3 and the fact|βi| ≤v⁻¹we have

|E^S13| ≤2€

E|s_n−E^sn|⁴(E|"i|⁴)²E|x_ii|⁴Š1/4

+v⁻¹€

E|s_n−E^sn|⁴(E|"i|⁴)²Š1/4€

E|x_ii|²I(|βi|>2)Š1/2

≤ C

nv. (16)

Therefore combining inequalities (13)-(16) we obtain

|S₁|=O 1

n

. (17)

Furthermore, we have the following expression forS₂, S₂=−1

n²

n

X

i=1

E(s_n−E^sn)γiβi

= a_n n²

Xn i=1

E(s_n−E^sn)γ_i− a²_n n²

Xn i=1

E(s_n−E^sn)γ_i"_iβ_i

=S₂₁+S₂₂+S₂₃+S₂₄+S₂₅, where

S₂₁= a_n n²

n

X

i=1

E(sn−n⁻¹t rD_i)γi, S₂₂=− a_n n^5/2

n

X

i=1

E(sn−E^sn)x_iiγiβi,

S₂₃= a_n n³

n

X

i=1

E(s_n−E^sn)βiγ²_i, S₂₄=−a_n n³

n

X

i=1

E(s_n−E^sn)γiβiξi,

S₂₅= a_n n²

Xn i=1

E|s_n−E^sn|²γiβi.

Here we will use the method which we used to handle the bound ofS₁. Firstly, we expressS₂₁ as follows

S₂₁= a_n n³

n

X

i=1

E((1+n⁻¹a^∗_iD⁻²_i a_i)βi)γi

=S₂₁₁+S₂₁₂,

where

S₂₁₁=−|a_n|² n⁴

Xn i=1

E(ˆγ_i)γ_i S₂₁₂= |a_n|² n³

Xn i=1

E((1+n⁻¹a^∗_iD⁻_i²a_i)β_i"_i)γ_i. From Lemma 4.1 and Hölder’s inequality we get

|S₂₁₁| ≤ C n⁴

n

X

i=1

€E|γˆi|²E|γi|²Š1/2

≤ C

n²v². Applying Lemma 4.2, Hölder’s inequality and (12), we obtain

S₂₁₂= |a_n|² n²

n

X

i=1

E(s_n−n⁻¹t rD⁻¹_i "i)γi

≤ C n³v

n

X

i=1

€E|"i|²E|γi|²Š1/2

≤ C

n²v². From the last two inequalities we obtain

|S₂₁|=O 1

n²v²

. (18)

ForS₂₂, we use Lemma 4.2 to get

|S₂₂| ≤ C n^7/2

Xn i=1

E|t rD⁻_i ¹−E^{t rD−}_i ¹||x_iiγ_iβ_i|+ C n^7/2v

Xn i=1

E|x_iiγ_iβ_i|. Notice thatx_ii andγi are independent. From Hölder’s inequality and Lemma 3.2 we have

E|x_iiγiβi| ≤CE|x_iiγi|+€

E|x_iiγi|²E|βiI(|βi|>2)|²Š_1/2

=O(n^1/2v^−1/2). Similarly we can get

E|t rD⁻_i¹−E^{t rD−}_i ¹||x_iiγiβi| ≤€

E|t rD⁻_i¹−E^{t rD−}_i ¹|⁴E|γi|⁴Š_1/4

=O(n^1/2v⁻²), which implies

|S₂₂| ≤ C

n²v². (19)

Now considerS₂₃. Using Lemma 4.2 again we obtain

|S₂₃| ≤ C n³

n

X

i=1

E|t rD⁻_i ¹−E^{t rD−}_i ¹||γ²_iβi|+ C n³v

n

X

i=1

E|γ²_iβi|

≤ C n³

Xn i=1

E|t rD⁻_i ¹−E^{t rD−}_i ¹||γ²_iβ_i|+ C n²v². Applying Lemma 3.2 and Hölder’s inequality we obtain

E|t rD⁻_i ¹−E^{t rD−}_i ¹||γ²_iβi| ≤2E|t rD⁻_i ¹−E^{t rD−}_i ¹||γ²_i| +€

(E|t rD⁻_i ¹−E^{t rD−}_i¹|²|γ_i|⁴)E|β_iI(|β_i|>2)|²Š1/2

≤€

E|t rD⁻_i¹−E^{t rD−}_i ¹|²|γi|⁴Š1/2

.

It follows from (11) that

E|t rD⁻_i ¹−E^{t rD−}_i ¹|²|γi|⁴

≤CE|t rD⁻_i ¹−E^{t rD−}_i¹|²€

v⁻²|t rD⁻_i ¹|²+n^1/2v⁻³|t rD⁻_i ¹|Š

≤C v⁻²E|t rD⁻_i ¹−E^{t rD−}_i ¹|⁴+C n²

v² E|t rD⁻_i ¹−E^{t rD−}_i ¹|²

≤C n²

v² E|t rD⁻_i ¹−E^{t rD−}_i ¹|²+ C v⁸

≤C n²

v² E|t rD⁻¹−E^{t rD−1}|²+ C v⁸. Then, we conclude that

|S₂₃| ≤ C nv

E^sn−E^sn

21/2

+ C

n²v². (20)

From Lemma 3.2, Lemma 4.2, Lemma 4.3 and Hölder’s inequality, it is easy to check that

|S₂₄| ≤ C n²v

Xn

i=1

€E|s_n−E^sn|⁴E|γi|⁴Š1/4€

E|βi|²I(|βi|>2)Š1/2

≤ C

n²v². (21) ForS₂₅, we use (5) to represent it in the form

S₂₅=− a²_n n²

Xn

i=1

E|s_n−E^sn|²γi+ a_n² n²

Xn

i=1

E|s_n−E^sn|²γi"iβi

=−S₂₅₁+S₂₅₂.

Using Lemma 4.3 and Hölder’s inequality we obtain

|S₂₅₁| ≤C n⁴

Xn

i=1

E|ξi−Eξi|²|γi|+ C n⁴

Xn

i=1

E|ξi−Eξi||t rD⁻¹_i −E^{t rD−1}_i ||γi|

≤ C

n^5/2v^5/2+ C

n^5/2v³ =O 1

n²v²

. (22)

Similarly we can obtain that E|s_n−E^sn|²γi"iβi≤€

E(|s_n−E^sn|⁸|γi|⁴)E|"i|⁴Š_1/4€

2+E|βi|²I(|βi|>2)Š_1/2

≤ C

n^1/2v^1/2

€E(|s_n−E^sn|⁸|γi|⁴)Š1/4

≤ C

n^1/2v^1/2

n⁻⁸E(|ξ_i−Eξ_i|⁸|γ_i|⁴) +n⁻⁸E(|t rD⁻_i ¹−E^{t rD−}_i ¹|⁸|γ_i|⁴)1/4

≤ C

n³v⁴. From the last inequality and (22) we obtain

|S₂₅| ≤ C

n²v². (23)

Combining inequalities (18)-(21) and (23), we conclude that, forv≥v₀

|S₂| ≤ C nv

E^sn−E^sn

21/2

+ C

n²v². (24)

From Lemma 3.2, Lemma 4.3 and Hölder’s inequality, we can check that

|S₃| ≤ C n²v

Xn

i=1

€E|s_n−E^sn|²(2+E|βi|²I(|βi|>2))Š1/2

≤ C n v

€E|s_n−E^sn|²Š1/2

. (25)

Therefore, it remians to get the bound ofS₄. Now we recall the equality (9), then we have S₄=a_nE|s_n−Es_n|²−a_n(S41+S₄₂+S₄₃+S₄₄),

and

S₄=−E^snE|s_n−E^sn|²−E|s_n−E^sn|²(s_n−E^sn). (26) Here

S₄₁= 1 n^3/2

Xn

i=1

E|s_n−E^sn|²x_iiβi, S₄₂=− 1 n²

Xn

i=1

E|s_n−E^sn|²γiβi,

S₄₃= 1 n²

n

X

i=1

E|s_n−E^sn|²ξiβi,

S₄₄=−E^snE|s_n−E^sn|²(s_n−E^sn)−E|s_n−E^sn|²(s_n−E^sn)². Comparing (26) withS₄₄, we obtain that

(1+a_nE^sn)E|s_n−E^sn|²(s_n−E^sn)

=−(an+E^sn)E|s_n−E^sn|²

+a_n(S₄₁+S₄₂+S₄₃−E|s_n−E^sn|²(s_n−E^sn)²), which implies that

−E|s_n−E^sn|²(s_n−E^sn)

=b_na⁻_n¹(an+E^sn)E|s_n−E^sn|²

−b_n(S₄₁+S₄₂+S₄₃−E|s_n−E^sn|²(s_n−E^sn)²), where b_n= (z+2E^sn(z))⁻¹. Thus denotingδ_n=n⁻¹Pn

i=1Eβ_i"_i, we conclude that S₄=(−E^sn+b_na⁻_n¹(an+E^sn))E|s_n−E^sn|²

−b_n(S₄₁+S₄₂+S₄₃−E|s_n−E^sn|²(s_n−E^sn)²)

=(a_n−δ_nb_nE^sn)E|s_n−E^sn|²

−b_n(S41+S₄₂+S₄₃−E|s_n−E^sn|²(sn−E^sn)²)

=(a_n+a_nδnb_n)E|s_n−E^sn|²

−b_n(δ²_nE|s_n−E^sn|²+S₄₁+S₄₂+S₄₃−E|s_n−E^sn|²(s_n−E^sn)²).

It is obvious thatS₄₂ andS₂₅ have the same bound. Using Lemma 3.2, Lemma 4.2 and Lemma 4.3 we get the following three inequalities

|E|s_n−E^sn|²(sn−E^sn)²| ≤E|s_n−E^sn|⁴≤ C n⁴v⁶,

|S₄₃| ≤ 1 nv

€E|s_n−E^sn|⁴Š1/2

≤ C

n³v⁴, and

|E|s_n−E^sn|²x_iiβi| ≤ |E|s_n−E^sn|²x_iia_n|+|E|s_n−E^sn|²x_iia_n"iβi|

≤ C

n²v³ +€

E|s_n−E^sn|¹⁶E|x_ii|⁸Š1/8€

E|"_i|⁴Š1/4€

2+E|β_i|²I(|β_i|>2)Š1/2

=O 1

n²v³

. Furthermore, from the definition ofδn and (12), we have

|δ_n|=

n⁻¹ Xn

i=1

€E^n−1D−_i ¹−E^sn+Eβ_i"²_iŠ

≤ C nv. Therefore, we obtain

S₄=a_nE|s_n−E^sn|²+O |b_n|

n²v²

, which combined with (17), (24) and (25) implies

|1−a²_n|E|s_n−E^sn|²≤ C₁|a_nb_n|

n +C₂|a_n| pn

€E|s_n−E^sn|²Š1/2

.

Then, from (6.91) and (6.95) in[10]which are under the existing fourth moment assumption, for 1>v>v₀,

|1−a²_n| ≥ |a_n(z+2s(z))|and|b_n| ≤2|z+2s(z)|⁻¹, we obtain the following inequality

E|s_n−E^sn|²≤ C₁

n|z+2s(z)|² + C₂ pn|z+2s(z)|

€E|s_n−E^sn|²Š1/2

. Solving this inequality, we obtain

E|s_n−E^sn|²≤ C n|z+2s(z)|², which complete the proof of the Lemma.

4 Basic lemmas

In this section we list some results which are needed in the proof.

Lemma 4.1. (Lemma B.26 of[6]) LetA be an n×n nonrandom matrix and X= (x1, . . . ,x_n)^∗ be a random vector of independent entries. Assume thatE^xi =0, E|x_i|² = 1, and E|x_j|^l ≤ νl. Then, for any p≥1,

E|X^∗AX−t rA|^p≤C_p

ν4t r(AA^∗)p/2

+ν2pt r(AA^∗)^p/2 , where C_pis a constant depending on p only.

Lemma 4.2. (Lemma 2.6 of[14]). Let z∈C^{+with v}=ℑz,AandBn×n withBHermitian,τ∈R^, andq∈C^N. Then

|t r((B−zI)⁻¹−(B+τqq^∗−zI)⁻¹)A| ≤ kAk v .

Lemma 4.3. (Lemma 8.7 of[6]) Under the assumption in Theorem 1.1, we have for any l≥1 E|s_n(z)−E^sn(z)|^2l≤ C

n^2lv^3l. (27)

Acknowledgement: The authors would like to thank the referee for his/her helpful suggestions.

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