getdoc41e2. 222KB Jun 04 2011 12:04:18 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 14 (2009), Paper no. 83, pages 2391–2417. Journal URL

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

CLT for Linear Spectral Statistics of Wigner matrices

∗

Zhidong Bai

†

Xiaoying Wang

‡

Wang Zhou

§

Abstract

In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order deriva-tives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.

Key words:Bernstein polynomial, central limit theorem, Stieltjes transform, Wigner matrices. AMS 2000 Subject Classification:Primary 15B52, 60F15; Secondary: 62H99.

Submitted to EJP on July 17, 2009, final version accepted October 14, 2009.

∗_{The first two authors were partially supported by CNSF 10871036 and NUS grant R-155-000-079-112 and the third} author was partially supported by grant R-155-000-076-112 at the National University of Singapore

†_{School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P.R.C. Email:} baizd@nenu.edu.cn

‡_{School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P.R.C. Email:} wangxy022@126.com

§_{Department of Statistics and Applied Probability, National University of Singapore,117546, Singapore. Email:} stazw@nus.edu.sg

1

Introduction and Main result

The random matrices theory originates from the development of quantum mechanics in the 1950’s. In quantum mechanics, the energy levels of a system are described by eigenvalues of a Hermitian operator on a Hilbert space. Hence physicists working on quantum mechanics are interested in the asymptotic behavior of the eigenvalues and from then on, the random matrices theory becomes a very popular topic among mathematicians, probabilitists and statisticians. The leading work, the famous semi-circle law for Wigner matrices was found in[19].

A real Wigner matrix of sizenis a real symmetric matrixW_n= (x_{i j})_1≤i,j≤nwhose upper-triangle en-tries(xi j)1≤i≤j≤nare independent, zero-mean real-valued random variables satisfying the following moment conditions:

(1)._∀i, E_|xii|2=σ2>0; (2).∀i< j, E|xi j|2=1. The set of these real Wigner matrices is called the Real Wigner Ensemble (RWE).

A complex Wigner matrix of size n is a Hermitian matrix W_n whose upper-triangle entries

(xi j)1≤i≤j≤n are independent, zero-mean complex-valued random variables satisfying the follow-ing moment conditions:

(1).∀i, E_|xii|2=σ2>0; (2).∀i< j, E|xi j|2=1, andExi j2 =0.

The set of these complex Wigner matrices is called the Complex Wigner Ensemble (CWE).

The empirical distribution F_n generated by the n eigenvalues of the normalized Wigner matrix

n−1/2Wn is called the empirical spectral distribution (ESD) of Wigner matrix. The semi-circle law states thatF_na.s. converges to the distributionF with the density

p(x) = 1

2π

p

4−x2_{, x ∈[−2, 2].}

Its various versions of convergence were later investigated. See, for example,[1],[2].

Clearly, one method of refining the above approximation is to establish the rate of convergence, which was studied in[3],[10],[12],[13],[18],[5]and[8]. Although the exact convergence rate remains unknown for Wigner matrices, Bai and Yao[6]proved that the spectral empirical process of Wigner matrices indexed by a set of functions analytic on an open domain of the complex plane including the support of the semi-circle law converges to a Gaussian process under fourth moment conditions.

To investigate the convergence rate of the ESD of Wigner matrix, one needs to use f as step func-tions. However, many evidences show that the empirical process associated with a step function can not converge in any metric space, see Chapter 9 of[8]. Naturally, one may ask whether it is possible to derive the convergence of the spectral empirical process of Wigner matrices indexed by a class of functions under as less assumption on the smoothness as possible. This may help us to have deeper understanding on the exact convergence rate of ESD to semi-circle law.

In this paper, we consider the empirical process of Wigner matrices, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law. More precisely, letC4(_U)denote the set of functions f :_{U →}Cthat has continuous

fourth-order derivatives, where_U is an open interval including the interval[₋2, 2], the support of

F(x). The empirical processGn¬{Gn(f)}indexed byC4(U)is given by

G_n(f)¬n

Z ∞

−∞

f(x)[F_n−F](d x), f ∈C4(_U) (1.1) In order to give a unified treatment for Wigner matrices, we define the parameterκwith values 1 and 2 for the complex and real Wigner matrices respectively. Also setβ=E(_|x12|2−1)2−κ. Our

main result is as follows.

Theorem 1.1. Suppose

E_|x_{i j|}6_≤M for all i,j. (1.2)

Then the spectral empirical process G_n =_{G_n(f): f _∈C4(_U)_}converges weakly in finite dimensions to a Gaussian process G:=_{G(f): f ∈C4(_U)_}with mean function

EG(f) =κ−1

4 [f(2) + f(−2)]−

κ−1

2 τ0(f) + (σ

2

−κ)τ2(f) +βτ4(f). and covariance function

c(f,g) ¬ E[_{G(f)₋EG(f)_}{G(g)₋EG(g)_}] = 1

4π2

Z 2

−2

Z 2

−2

f′(t)g′(s)V(t,s)d t ds

where

V(t.s) =

σ2−κ+1

2βts

p

(4₋t2_)(4−s2₎

+κlog

 

4₋ts+p(4₋t2_)(4−s2₎

4−ts−p(4−t2)(4−s2)

 

and

τl(f) = 1 2π

Z π

−π

f(2cosθ)eilθdθ= 1

2π

Z π

−π

f(2cosθ)cos(lθ)dθ

= 1

π

Z 1

−1

f(2t)T_l(t)_p 1

1₋t2 d t.

Here_{T_l,l _≥0_}is the family of Chebyshev polynomials.

Remark 1.2. In the definition ofG_n(f),θ =R f(x)d F(x)can be regarded as a population param-eter. The linear spectral statisticθˆ=R f(x)d F_n(x)is then an estimator ofθ. We remind the reader that the centerθ = R f(x)d F(x), instead ofER f(x)d Fn(x), has its strong statistical meaning in the application of Theorem 1.1. For example, in quantum mechanics, Wigner matrix is a discretized version of a random linear transformation in a Hilbert space, and semicircular law is derived under

ideal assumptions. Therefore, the quantum physicists may want to test the validity of the ideal as-sumptions. Therefore, they may suitably select one or several f’s so that θ’s may characterize the semicircular law. Using the limiting distribution of G_n(f) =n( ˆθ −θ), one may perform statistical test of the ideal hypothesis. Obviously, one can not apply the limiting distribution ofn( ˆθ−E_θˆ)to the above test.

Remark 1.3. Checking the proof of Theorem 1.1, one finds that the proof still holds under the assumption of bounded fourth moments of the off-diagonal entries is finite, if the approximation

G_n(f_m) ofG_n(f) is of the desired order. Thus, the assumption of 6-th moment is only needed in deriving the convergence rate of_kF_n−F_k. Furthermore, the assumption of the fourth derivative of f is also related to the convergence rate of kFn−Fk. If the convergence rate of kFn −Fk is improved and/or proved under weaker conditions, then the result of Theorem 1.1 would hold un-der the weakened conditions. We conjecture that the result would be true if we only assume the fourth moments of the off-diagonal elements of Wigner matrices are bounded and f have the first continuous derivative.

Pastur and Lytova in[17]studied asymptotic distributions ofnR f(x)d(F_n(x)₋EF_n(x))under the conditions that the fourth cumulant of off-diagonal elements of Wigner matrices being zero and the Fourier transform of f(x) having 5th moment, which implies that f has the fifth continuous derivative. Moreover, they assume f is defined on the whole real line. These are stronger than our conditions.

Remark 1.4. The strategy of the proof is to use Bernstein polynomials to approximate functions in

C4(_U). This will be done in Section 2. Then the problem is reduced to the analytic case, which has been intensively discussed in Bai and Yao[6]. But the functions in[6]are independent ofnand thus one can choose fixed contour and then prove that the Stieltjes transforms tend to a limiting process. In the present case, the Berstein polynomials depend onnthrough increasing degrees and thus they are not uniformly bounded on any fixed contour. Therefore, we cannot simply use the results of Bai and Yao[6]and thus we have to choose a sequence of contours approaching to the real axes so that the approximating polynomials are uniformly bounded on the corresponding contours. On this sequence of contours, the Stieltjes transforms don’t have a limiting process. Our Theorem 1.1 cannot follow directly from[6]. We have to find alternative ways to prove the CLT.

Remark 1.5. It has been also found in literature that the so-called secondary freeness is proposed and investigated in free probability. Readers of interest are referred to Mingo [14; 15; 16]. We shall not be much involved in this direction in the present paper. As a matter, we only comment here that both the freeness and the secondary freeness are defined on sequences of random matrices (or ensembles) for which the limits of expectations of the normalized traces of all powers of the random matrices and the limits of covariances of unnormalized traces of powers of the random matrices exist. Therefore, for a single sequence of random matrices, the existence of these limits has to be verified and the verification is basically equivalent to moment convergence method. Results obtained in[14]is in some sense equivalent to those of[17].

The paper is organized as follows. The truncation and re-normalization step is in Section 3. We derive the mean function of the limiting process in Section 4. The convergence of the empirical processes is proved in Section 5.

2

Bernstein polynomial approximation

It is well-known that if ˜f(y)is a continuous function on the interval[0, 1], the Bernstein polynomials ˜

f_m(y) =

m

X

k=0

_m

k

yk(1₋y)m−kf˜

_k

m

converge to ˜f(y)uniformly on[0, 1]asm_{→ ∞}. Suppose ˜f(y)_∈C4[0, 1]. Taylor expansion gives ˜

f

_k

m

= ˜f(y) +

_k

m− y

˜

f′(y) +1

2

_k

m−y

2

˜

f′′(y) +1

3!

_k

m−y

3

˜

f(3)(y) + 1

4!

_k

m−y

4

˜

f(4)(ξy).

Hence

˜

fm(y)−f˜(y) =

y(1₋y)f˜′′(y)

2m +O

₁

m2

. (2.1)

For the functionf _∈C4(_U), there existsa>2 such that[₋a,a]_{⊂ U}. Make a linear transformation

y = K x+ 1

2, ε ∈(0, 1/2), whereK = (1−2ε)/(2a), then y ∈[ε, 1−ε]if x ∈[−a,a]. Define

˜

f(y)¬ f(K−1(y₋1/2)) = f(x), y _∈[ε, 1₋ε], and define

f_m(x)¬ ˜f_m(y) =

m

X

k=1

_m

k

yk(1−y)m−kf˜

_k

m

.

From (2.1), we have

f_m(x)₋f(x) = f˜_m(y)₋ ˜f(y) = y(1− y)

˜

f′′(y)

2m +O

₁

m2

.

Since ˜h(y)¬ y(1₋ y)f˜′′(y) has a second-order derivative, we can use Bernstein polynomial ap-proximations once again to get

˜

hm(y)−˜h(y) = m

X

k=1

_m

k

yk(1− y)m−k˜h(k

m)−˜h(y)

= O

₁

m

. So, withh_m(x) =˜h_m(K x+1

2),

f(x) = fm(x)− 1

2mhm(x) +O

₁

m2

Therefore,G_n(f)can be split into three parts:

G_n(f) = n

Z ∞

−∞

f(x)[F_n−F](d x)

= n

Z

f_m(x)[F_n−F](d x)₋ n

2m

Z

h_m(x)[F_n−F](d x) +n

Z

f(x)₋fm(x) + 1 2mhm(x)

[Fn−F](d x)

= ∆₁+ ∆₂+ ∆₃. For∆₃, by Lemma 6.1 given in Appendix,

kFn−Fk=Op(n−2/5).

Here and in the sequel, the notationZn =Op(cn) means that for anyε > 0, there exists anM >0 such that sup_nP(_|Z_{n| ≤} M c_n) < ε. Similarly, Z_n = o_p(c_n) means that for any ε >0, lim_nP(_|Z_{n| ≤}

εc_n) =0.

Takingm2= [n3/5+ε0_{], for someε}

0>0 and using integration by parts, we have that

∆₃ = ₋n

Z

f(x)₋f_m(x) + 1

2mhm(x)

′

F_n(x)₋F(x)

d x

= Op(n−ε0) since f(x)₋ fm(x) +₂1_mhm(x)

′

=O(m−2). From now on we choose ε0 = 1/20, and thenm= n13/40.

Note that f_m(x)andh_m(x)are both analytic. Following from the result proved in Section 5, replac-ing f_m byh_m, we obtain that

∆₂=O(∆1)

m =op(1).

It suffices to consider ∆₁ = Gn(fm). In the above, fm(x) and gm(y) are only defined on the real line. Clearly, the two polynomials can be considered as analytic functions on the complex regions

[₋a,a]_×[₋ξ,ξ]and[ε, 1₋ε]_×[₋Kξ,Kξ], respectively.

Since g_∈C4[0, 1], there is a constant M, such that_|g(y)_|< M, _∀y _∈[ε, 1₋ε]. Noting that for

(u,v)_∈[ε, 1₋ε]_×[₋Kξ,Kξ],

|u+i v_|+_|1₋u₋i v_| = pu2_+v2₊p_(1−u)2_+v2

≤ u

–

1+ v

2

2u2

™

+ (1₋u)

–

1+ v

2

2(1₋u)2

™

≤ 1+ v

2

ε

we have, for y=K x+1/2=u+i v,

|f_m(x)_| = _|f˜_m(y)_|=

n

X

k=1

_m

k

yk(1₋y)m−kf˜

_k

m

≤ M

‚

1+ v

2

ε

Œm

If we take _|ξ| ≤ K/pm, then _|f˜m(y)| ≤ M

€

1+K2/(mε)Šm _→ M eK2/ε, as m → ∞. So ˜fm(y) is bounded when y ∈ [ε, 1−ε]_×[₋K/pm,K/pm]. In other words f_m(x) is bounded when x ∈

[₋a,a]_×[₋1/pm, 1/pm]. Let γm be the contour formed by the boundary of the rectangle with vertices(_±a±i/pm). Similarly, we can show thathm(x), fm′(x)andh′m(x)are bounded onγm.

3

Simplification by truncation and Preliminary formulae

3.1

Truncation

As proposed in [9], to control the fluctuations around the extreme eigenvalues under condi-tion (1.2), we will truncate the variables at a convenient level without changing their weak limit. Condition (1.2) implies the existence of a sequenceηn↓0 such that

1

n2η4

n

X

i,j

E

h

|xi j|4I_|x i j|≥ηnpn

i

=o(1).

We first truncate the variables as ˆxi j = xi jI|xi j|≤ηnpn and normalize them to ˜xi j = ( ˆxi j−Eˆxi j)/si j,

wheres_{i j} is the standard deviation of ˆx_{i j}.

LetFˆ_n and ˜F_ndenote the ESDs of the Wigner matricesn−1/2( ˆx_{i j})andn−1/2(x˜_{i j}), andGˆ_nand ˜G_nthe corresponding empirical process, respectively. First of all, by (1.2),

P(G_n6= ˆG_n) ¶ P(F_n= ˆ₆ F_n)¶P( ˆx_{i j6}= ˜x_{i j})

¶ X

i,j

P(_|x_{i j| ≥}ηn p

n)

¶ (ηn p

n)−4X

i,j

E_|x_{i j|}4I_|x

i j|≥ηnpn=o(1).

Second, we compare ˜G_n(f_m)andGˆ_n(f_m). Note that max

i,j |1−si j|

¶ max i,j |1−s

2

i j|

= max i,j

h

E(_|x_{i j|}2I_|x

i j|≥ηnpn) +|E(xi jI|xi j|≥ηnpn)| 2i

≤ (n−1η−_n2+Mη−_n6n−3)max i,j [

E(_|x_{i j|}4I_|x i j|≥ηn

p

n)]→0. Therefore, there exist positive constantsM₁ andM₂so that

X

i,j

E(_|xi j|2|1−s−i j1|

2₎

≤M1

X

i,j

(1−s2_{i j})2_≤ M2

η4

nn2

X

i,j

E(x_{i j}4I_|x

Since f(x)_∈C4(_U)implies that f_m(x)are uniformly bounded in x_∈[₋a,a]andm, we obtain

E_|G˜_n(f_m)₋Gˆ_n(f_m)_|2 _≤ ME

  

n

X

j=1

|λ˜n j−λˆn j|

  

2

≤M nE

n

X

j=1

|λ˜n j−λˆn j|2

= M nEX

i,j

|n−1/2(x˜_{i j}−ˆx_{i j})_|2

≤ M

  

X

i,j

(E_|x_{i j|}2)_|1₋s−_{i j}1_|2+X

i,j

|E( ˆx_{i j})_|2s−_{i j}2

  

= o(1),

where ˜λn j andλˆn j are the jth largest eigenvalues of the Wigner matricesn−1/2(˜xi j)andn−1/2( ˆxi j), respectively.

Therefore, the weak limit of the variables(Gn(fm))is not affected if we substitute the normalized truncated variables ˜x_{i j} for the original x_{i j}.

From the normalization, the variables ˜x_{i j} all have mean 0 and the same absolute second moments as the original variables. But for the CWE, the conditionEx_{i j}2 =0 does no longer remain true after these simplifications. Fortunately, we have the estimateE˜x_{i j}2 = o€n−1η2_nŠ, which is good enough for our purposes.

For brevity, in the sequel we still use xi j to denote the truncated and normalized variables ˜xi j.

3.2

Preliminary formulae

Recall that for a distribution functionH, its Stieltjes transforms_H(z)is defined by

sH(z) =

Z

1

x₋zd H(x), z∈C. The Stieltjes transforms(z)of the semicircle law F is given bys(z) =₋1

2(z−sgn(Im(z))

p

z2₋₄₎

forz withIm_{(z) 6=0 which satisfies the equations(z)}2_{+zs(z) +1=0. Here and throughout the}

paper,pzof a complex numberzdenotes the square root ofzwith positive imaginary part.

Define D= (n−1/2W_n−z I_n)−1. Letαk be the kth column ofWn with xkk removed andWn(k)the submatrix extracted fromW_n by removing itskth row andkth column. DefineD_k= (n−1/2W_n(k)₋

following notations:

βk(z) = − 1 p

nxkk+z+n

−1_α∗

kDkαk,

δn(z) = − 1

n

n

X

j=1

ǫk(z)

βk(z) ,

ǫk(z) = 1 p

nxkk−

1

nα

∗

kDkαk+Esn(z),

q_k(z) = _p1

nxkk−

1

n

”

α∗_kD_k(z)αk−t r Dk(z)

—

,

ω_k(z) = _−p1

nxkk+n

−1_α∗

kDkαk−s(z),

b_n(z) = n[Es_n(z)₋s(z)]. The Stieltjes transforms_n(z)ofF_nhas the representation

sn(z) = 1

nt r D=

1

nt r

_W

n p

n−z In

−1

= ₋1

n

n

X

k=1

1

βk(z)

=₋ 1

z+Es_n(z)+

δn(z)

(z+Es_n(z)). (3.2)

Throughout this paper,M may denote different constants on different occassions andε_na sequence of numbers which converges to 0 asngoes to infinity.

4

The mean function

Letλe x t(n−1/2Wn)denote both the smallest and the largest eigenvalue of the matrixn−1/2Wn (de-fined by the truncated and normalized variables). Forη0<(a−2)/2, define

A_n=_{|λ_{e x t}(n−1/2W_n)_{| ≤}2+η0}.

Then, by Cauchy integral we have

∆₁=G_n(f_m) = 1

2πi

Z I

γm f_m(z)

z₋xn[Fn−F](d x)dz IAn

+

Z

f_m(x)n[F_n−F](d x)I_Ac n.

SinceP(Ac_n) =o(n−t)for anyt>0 (see the proof of Theorem 2.12 in[4]),

Z

f_m(x)n[F_n−F](d x)I_Ac n→0

in probability. It suffices to consider

∆ = ₋ 1

2πi

I

γm

fm(z)n[sn(z)−s(z)]IAndz. (4.3)

In the remainder of this section, we will handle the asymptotic mean function of∆. The convergence of random part of∆will be given in Section 5.

To this end, writeb_n(z) =n[Es_n(z)₋s(z)]andb(z)¬[1+s′(z)]s3(z)[σ2_{−1+(κ−1)s}′_(z)+βs2_(z)].

Then we prove

E∆_h = ₋ 1

2πi

I

γmh

fm(z)b(z)dz

− 1 2πi

I

γmh

fm(z)[n(Esn(z)IAn−s(z))−b(z)]dz+o(1)

¬ R₁+R₂+o(1) (4.4)

∆_v = ₋ 1

2πi

I

γmv

fm(z)n[sn(z)−s(z)]IAndz=op(1), (4.5)

where and in the sequelγmhdenotes the union of the two horizontal parts ofγmandγmvthe union of the two vertical parts. The limit ofR1 is given in the following proposition.

Proposition 4.1. R1 tends to

EG(f) = κ−1

4 [f(2) + f(−2)]−

κ−1

2 τ0(f) + (σ

2

−κ)τ2(f) +βτ4(f) for both the CWE and RWE.

Proof. Since fm(z)are analytic functions, by[6], we have

R₁ = ₋ 1

2πi

I

γmh

f_m(z)b(z)dz

≃ κ−1

4 [fm(2) +fm(−2)]−

κ−1

2 τ0(fm) + (σ

2

−κ)τ2(fm) +βτ4(fm), wherea≃ bstands for a/b→1 asn→ ∞,

τl(fm) = 1 2π

Z π

−π

f_m(2cosθ)eilθdθ.

As fm(t)→ f(t)uniformly ont∈[−a,a]as m→ ∞, it follows that

R1−→

κ−1

4 [f(2) + f(−2)]−

κ−1

2 τ0(f) + (σ

2

Since f_m(z)is bounded on and inγm, in order to prove R2→0 asm→ ∞, it is sufficient to show

that

|b_nA(z)₋b(z)_|=o(1) uniformly on γ_mh, (4.6) where b_nA=n(Es_n(z)I_A

n−s(z)).

We first consider the casez∈γ_mh. Since

nEδn(z) = −E n

X

k=1

ǫk(z)

βk(z)

=₋E

n

X

k=1

ǫk(z)

z+Es_n(z)₋ǫk(z) ,

by the identity

1

u₋ε =

1

u

‚

1+ε

u+...+

εp

up +

εp+1

up_(u−ε)

Œ

, we have

nE_δn(z) = ₋E

n

X

k=1

ǫk(z)

z+Esn(z)−

E

n

X

k=1

ǫ2_k(z) [z+Es_n(z)]2

−E

n

X

k=1

ǫ_k3(z) [z+Esn(z)]3

−E

n

X

k=1

ǫ4_k(z) [z+Esn(z)]3βk(z)

¬ S1+S2+S3+S4.

Now supposez∈γ_mh. In order to analyzeS₁,S₂,S₃ andS₄, we present some facts.

Fact 1.

|Es_n(z)₋s(z)_| =

1

nEt r D(z)−s(z)

=

Z

1

x−z(EFn−F)(d x)

¶ 1

vkEFn−Fk=O

₁

p

nv

=O(n−2780).

where we have used Lemma 6.1. This implies

|Es_n(z)_|¶_|Es_n(z)₋s(z)_|+_|s(z)_|¶O(_p1

nv) +M ¶M. Fact 2.

E_|ǫk(z)|4 = E

1 p

nxkk−

1

nα

∗

kDk(z)αk+Esn(z)

4

¶ M[ 1

n2E|xkk| 4₊ 1

n4E|α

∗

kDk(z)αk−t r Dk(z)|4

+ 1

n4E|t r Dk(z)−t r D(z)| 4₊ 1

n4E|t r D(z)−Et r D(z)| 4_]

≤ M

n2_v4 =M n

where we have used_|t r D(z)₋t r D_k(z)_|¶1/v,E_|t r D(z)₋E t r D(z)_|2l¶cn−2lv−4l(∆ +v)l (l _≥1)

which follows from the proof of (4.3) in[7], and

E_|α∗_kDk(z)αk−t r Dk(z)|4 ¶ M

(E_|x12|4t r Dk(z)D∗k(z))

2

+nηnE|x12|6t r(Dk(z)D∗k(z))

2 ¶ M – n v2 ‹2 + n 2_η n v4 ™

= M n

2 v4

which is derived from Lemma 6.2.

Fact 3.

From (3.2) we have

Esn(z) =

−z+pz2_−4+4Eδ

n(z)

2 , (4.7)

where

|Eδn(z)| = | 1

nE

n

X

k=1

ǫk(z)

βk(z) |¶ 1

nv

n

X

k=1

E_|ǫk(z)|¶ 1

nv

n

X

k=1

(E_|ǫk(z)|4)

1 4

¶ _pM

nv2 =M n

−407.

Fact 4.

1

z+Es_n(z)

=

Es_n(z)

1₋E_δn(z)

¶M.

1

z+Es_n(z)−

1

z+s(z)

¶ M|Es_n(z)₋s(z)_|

¶ _pM

nv =M n

−2780.

Now we can estimateS₁,S₂,S₃ andS₄. First, we proveS_4→0. By_|βk(z)|−1¶v−1, Facts 2 and 4, |S4|=

E n X

k=1

ǫ_k4(z) [z+Esn(z)]3βk(z)

¶ n X

k=1 M

v E|ǫk(z)| 4₌ M

nv5 =M n

−₁₆3_.

Similarly, forS3, we have

|S_3|=

E n X

k=1

ǫ3_k(z) [z+Es_n(z)]3

¶M n X

k=1

|E_ǫk(z)_|3=M n_· 1

n3/2v3 =M n

−1 80.

ForS1, we will prove that

S1=s2(z)[1+s′(z)]|+o(1), uniformly on γmh. By (4.7) and Fact 3, we have

1

z+Es_n(z) =

z−pz2−4(1−Eδ_n(z))

where theo(1)is uniform forz_∈γmh. Thus, we have

S₁ = ₋

n

X

k=1

Eǫk(z)

z+Esn(z)

= 1

n

n

X

k=1 E

1+ 1

nα

∗

kD

2

k(z)αk

βk(z+Esn(z))

= 1

n

n

X

k=1

1+1_nEt r D2_k(z) (z+Es_n(z))2 +

1

n

n

X

k=1 E

(1+1_nα∗_kD2_k(z)αk)ǫk(z)

βk(z+Esn(z))2

= S₁₁+S₁₂. Using the fact

1+1

nα

∗

kD

2

k(z)αk

β_k(z)

≤ 1 v,

and Fact 2, forz_∈γmh, we obtain |S_{12| ≤} M

nv

n

X

k=1

E_|ǫk(z)_{| ≤} M

nv·n·n

−1/2_v−1

≤M n−7/20. (4.9)

To estimateS11, we use Lemma 6.1 and the fact thatk(n−1)Fnk−1−nFnk ≤1. For allz∈γmh,

1

nEt r D 2

k(z)−s′(z)

=

Z ₍n−1

n EF k

n−1−F)(d x)

(x−z)2

¶M v−2_kn−1 n EF

k

n−1−Fk ≤M v− 2_n−1/2

≤M n−7/40. (4.10) By (4.8) and (4.10), for allz∈γmh,

|S₁₁₋s2(z)(1+s′(z))_|=o(1). Finally we deal with

S2=−

n

X

k=1

E_ǫ2

k(z)

[z+Esn(z)]2

=₋s2(z)(1+o(1))

n

X

k=1

Eǫ_k2(z).

By the previous estimate forEǫk(z), it follows that |Eǫk(z)|2 =

1

n2

E(t r D(z)₋t r Dk(z))

2

¶ M

n2_v. Eǫ2_k(z) = E[ǫk(z)−Eǫk(z)]2+O(

1

n2v)

= E

h 1

p

nxkk−

1

n[α

∗

kDk(z)αk−t r Dk(z)] −1

n[t r Dk(z)−Et r Dk(z)]

i2

+O( 1

n2_v)

= σ

2 n +

1

n2E[α

∗

kDk(z)αk−t r Dk(z)]2 (4.11) − 1

n2E[t r Dk(z)−Et r Dk(z)]

2_+O( 1 n2_v).

By the proof of (4.3) in[7], we have 1

n2E|t r Dk(z)−Et r Dk(z)|

2_¶ M n2v4

₁

p

n+v

= M

n2v3.

Hence we neglect this item. In order to estimate the expectation in (4.11), we introduce the notationαk= (x1k,x2k, ...,xk−1k,xk+1k, ...,xnk)∗¬(xi)and Dk(z)¬(di j). Note thatαk andDk(z) are independent.

E[α∗_kDk(z)αk−t r Dk(z)]2

= E

  

X

i6=j

d_{i j}¯x_ix_j+X

i

d_ii(_|x_i|2−1)

   2 = E    X

i6=j

X

s6=t

d_{i j}d_st¯x_ix_jx¯_sx_t

  +E

 

X

i

d_ii2(_|x_i|2−1)2

 

= E(¯x1)2E(x2)2E

  

X

i6=j

d_{i j}2

 

+E|x1|

2_E

|x2|2E

  

X

i6=j

d_{i j}d_ji

  

+E(_|x1|2−1)2E

 

X

i

d_ii2

 .

Here we need to consider the difference between the CWE and RWE.

For the RWE, all the original and truncated variables have the properties: Ex_{i j} = 0 andE_|x_{i j|}2 =

Ex_{i j}2 =1. So

E[α∗_kD_k(z)αk−t r Dk(z)]2 = 2E

  

X

i6=j

d_{i j}2

 

+E(|x1|

2

−1)2E

 

X

i

d_ii2

 

= 2E

  

X

i,j

d_{i j}2

 

+ [E(|x1|

2

−1)2₋2]E

 

X

i

d_ii2

 .

For the CWE, the truncated variables have the properties: Exi j = 0,E|xi j|2 = 1 and Ex2i j =

o€n−1η−_n2Š. Note that_|P i,j

d_{i j}2_|¶t r(D_k(z)D∗_k(z))¶n/v2andP_i|d_ii2_|¶n/v2. So

E[α∗_kD_k(z)αk−t r Dk(z)]2

= E    X i j

d_{i j}d_ji

  +

”

E(_|x_1|2₋1)2₋1—E

 

X

i

d_ii2

 +o

‚

1

nv2_η4

n

Œ

.

We introduce the parametersκwith values 1 and 2 for the CWE and RWE, andβ=E(_|x_1|2₋1)2₋κ, which allow us to have the following unified expression,

E[α∗_kDk(z)αk−t r Dk(z)]2 = κE

  

X

i j

di jdji

  +βE

 

X

i

d_ii2

 +o

‚

1

nv2η4

n

Œ

Hence

Eǫ2_k(z) = σ

2 n +

1

n2



κE[t r D2_k(z)] +βE  

X

i

d_ii2

 +o

‚

1

nv2η4

n

Œ +

M n2v3,

which combined with

E

1

nt r D 2

k(z)−s′(z)

2

¶ 1

v4(kF

k

n−1−Fnk+kFn−Fk)2

¶ M

v4

‚

1

n+

1

n25−η

Œ2

¶M n−203+2η

and

E_|d_ii2₋s2(z)_| ¶ E_|[D_k(z)]_ii+s(z)_||[D_k(z)]_ii−s(z)_|

¶ M

₁

v +1

E_|[D_k(z)]_ii−s(z)_|¶ M v

”

E_|[D_k(z)]_ii−s(z)_|2—1/2

¶ M

v

₁

nv4

1

2

=_pM

nv3,

gives

|S₂+s2(z)[σ2+κs′(z) +βs2(z)]_|¶ _pM nv3.

Summarizing the estimates ofS_i,i=1, ..., 4, we have obtained that |nEδn(z) +s2(z)[σ2−1+ (κ−1)s′(z) +βs2(z)]|¶

M

p

nv3

Note that

b_n(z) = n[Es_n(z)₋s(z)] = 2sgn(

Im_(z))nEδn(z)

p

z2_−4(1−Eδ

n(z)) +

p

z2₋₄

= sgn(

Im_(z))nEδn(z)

p

z2₋₄

(1+o(1)).

and that

s′(z) =₋ s(z)

z+2s(z) =−

s(z)

sgn(Im_(z))p_z2₋₄

. The second equation is equivalent to

sgn(_ℑ(z))

p

z2₋₄

=s(z)(1+s′(z)).

From the above two equations, we conclude

Then, (4.6) is proved by noticing that_|b_n(z)₋b_nA(z)_{| ≤}nv−1P(Ac_n)_≤o(n−t)for any fixed t. Now, we proceed the proof of (4.5). We shall find a setQn such thatP(Qnc) =o(n−1)and

I

γmv

f_m(z)n(s_n(z)I_Q

n−s(z))dz=op(1). (4.12)

By continuity of s(z), there are positive constants M_l and M_u such that for all z _∈ γmv, Ml ≤ |z+s(z)_{| ≤}M_u. DefineB_n=_{min

k |βk(z)|

I_A

n> ε}, whereε=Ml/4. So P(B_nc) = P(min

k |βk(z)|

I_A

n¶ε)¶

n

X

k=1

P(_|βk(z)|IAn¶ε)

¶

n

X

k=1

P(_|βk(z)−z−s(z)|IAn¾ε) +nP(A

c n)

¶ 1

ε4

n

X

k=1 E

|βk(z)−z−s(z)|4IAn

+nP(Ac_n)

¶ Mh 1 n2E|xkk|

4₊ 1 n4E

|α∗_kDk(z)αk−t r Dk(z)|4IAn

+E

|1

nt r Dk(z)−s(z)| 4_I

An

i

+nP(Ac_n).

LetA_nk=_{|λ_{e x t}(n−1/2W_nk)_{| ≤}2+η₀}. It is trivially obtained thatA_n⊆A_nk(see[11]). Noting the independence ofI_A

nk andαk, we have n−4E

|α∗_kDk(z)αk−t r Dk(z)|4IAn

≤ n−4E

[Ek_|α∗

kDk(z)αk−t r Dk(z)|4]IAnk

≤ M n−4E

[ E_|x_12|4t r D_k(z)D∗_k(z)2

+E_|x_12|8t r D_k(z)D∗_k(z)2

]I_A nk

≤ M n−2,

whereEkdenotes the expectation taken only aboutαk. Therefore

P(Bc_n)_≤M n−2+n−2+ (n−2/5+η)4

+nP(Ac_n)_≤M n−8/5+4η.

This givesP(Bc_n) =o(n−1)as n→ ∞. DefineQn¬An∩Bn, we haveP(Qn)→1, asn→ ∞. Similar to (4.7), we have forz_∈γmv,

Es_n(z)I_Q n=−

P(Q_n)

z+Es_n(z)I_Q n

+ δnQ(z)

z+Es_n(z)I_Q n

, (4.13)

where

δnQ(z) = 1

n

n

X

k=1 EǫkQ

(z)I_Q n

βk(z)

ǫkQ(z) = 1

p_nx_kk− 1 nα

∗

Similar to (4.7), we have

Es_n(z)I_Q n=−

1 2

z₋sgn(Im_(z))

Æ

z2_−4(P(Q

n)−δnQ(z))

. Therefore,

bnQ(z) = n(Esn(z)IQn−s(z))

= 2n[P(Q

c

n) +δnQ(z)]

sgn(Im_(z))[p_z2_{−4(P(Qn)−δnQ(z)) +}p_z2_−4]

.

We shall prove that bnQ(z)is uniformly bounded for all z∈γmv. Noticing that|z2−4|>(a−2)2 and the factP(Qc_n) =o(1), we only need to show that

nδnQ(z) is uniformly bounded forz∈γmv. (4.14) Rewrite

nδnQ(z) = n

X

k=1

EǫkQIQn z+Es_n(z)I_Q

n

+

n

X

k=1

E_ǫ2

kQIQn

βk(z)[z+Esn(z)IQn]

.

At first we have

E_ǫkQI_A nk =

1

nE

h

t r D(z)I_Q

nP(Ank)−t r Dk(z)IAnk

i

= 1

nE

_I

Qn

βk

P(A_nk)₋t r D_k(z)(I_Qc

nAnk+IQnP(A

c nk))

=O(1/n).

Here, the result follows from facts that 1/βk is uniformly bounded when Qn happens, 1_nDk(z) is bounded whenAnk happens and P(Qcn) = o(1/n). From this and the facts that Esn(z)IQn → s(z)

uniformly onγmv and

E_|ǫkQ|I_Qc nAnk ≤

1

nE

h

t r D(z)I_Q nP(Q

c

nAnk)−t r Dk(z)IQc nAnk

i

= O(1/n),

we conclude that the first term in the expansion ofnδnQ(z)is bounded.

By similar argument, one can prove that the second term of the expansion ofnδnQ(z)is uniformly bounded onγmv. Therefore, we have

Proposition 4.2. H_γ

mv fm

(z)n(Esn(z)IQn−s(z))dz→0in probability as n→ ∞.

Therefore, to complete the proof of (4.5), we only need to show that

Proposition 4.3. H γmv fm

(z)n(sn(z)IQn−Esn(z)IQn)dz→0in probability as n→ ∞.

5

Convergence of

∆

₋

E

∆

Let_Fk= (x_{i j},k+1_≤i,j_≤n)for 0_≤k_≤nandE_k(_·) =E(_·|Fk). Based on the decreasing filtration

(_Fk), we have the following well known martingale decomposition

n[sn(z)−Esn(z)] =t r D(z)−Et r D(z)

=

n

X

k=1

(E_k−1−E_k)(t r D(z)₋t r D_k(z)) =

n

X

k=1

(E_k−1−E_k) 1+n

−1_α∗

kD

2

k(z)αk

n−1/2_x

kk−z−n−1α∗kDk(z)αk

=₋

n

X

k=1

(E_k−1−E_k) d

dzlog

₁

p

nxkk−z−

1

nα

∗

kDk(z)αk

. This decomposition gives

∆₋E∆ = 1

2πi

n

X

k=1

(E_k−1−E_k)

I

γm fm(z)

d dzlog

₁

p

nxkk−z−

1

nα

∗

kDk(z)αk

dz

= ₋ 1

2πi

n

X

k=1

(E_k−1−E_k)

I

γm

f_m′(z)log

₁

p

nxkk−z−

1

nα

∗

kDk(z)αk

dz

= ₋ 1

2πi

n

X

k=1

(E_k−1−E_k)

I

γm

f_m′(z)log

1+ qk(z)

−z−n−1t r Dk(z)

dz

= R₃₁+R₃₂+R₃₃

whereq_k(z) = p1_nx_kk−1_n”α∗_kD_k(z)αk−t r Dk(z)

—

and

R₃₁ = ₋ 1

2πi

n

X

k=1

(E_k−1−E_k)

I

γmh

f_m′(z)log

1+ qk(z)

−z₋n−1_{t r D}

k(z)

dz

R32 = −

1 2πi

n

X

k=1 I_A

nk(Ek−1−Ek)

I

γmv

f_m′(z)log

1+ qk(z)

−z−n−1t r D_k(z)

dz

R33 = −

1 2πi

n

X

k=1 I_Ac

nk(Ek−1−Ek)

I

γmv

f_m′(z)log

1+ qk(z)

−z−n−1t r D_k(z)

dz. At first, we note that

P(R336=0)≤P(

[

Ac_nk)_≤P(Ac_n)_→0. Next, we note thatR₃₂is a sum of martingale differences. Thus, we have

E_|R32|2 ≤

kf_m′k2n−1340

π2

n

X

k=1

sup z∈γmv

EI_A nk

q_k(z)

z+n−1t r Dk(z)

2

= kf

′

mk

2

π2_n53₄₀

n

X

k=1

sup z∈γmv

EI_A nk

σ2+2

nt r Dk(z)Dk(z¯)

z+n−1_{t r D}

k(z)

2 . (5.15)

since

f

_m′

(t

)

_→

f

′

(t)

,

g

_m′

(t

)

_→

g

′

(t)

uniformly on

[

₋

2, 2

]

.

For

Q

₂

, since

s(z

₁

)s(z

₂

)

f

_m′

(z

₁

)g

_m′

(z

₂

)

is bounded on

γ

mh

×

γ

mh

, in order to prove

Q

2

→

0 in

probabil-ity, it is sufficient to prove that

Γ

_n

(z

₁

,

z

2

)

converges in probability to

Γ(z

1

,

z

2

)

uniformly on

γ

mh×

γ

mh

.

Decompose

Γ

_n

(z

₁

,

z

₂

)

as

Γ

_n

(z

₁

,

z

2

)

=

n

X

k=1

E

_k

[E

_k−1

q

k

(z

1

)E

k−1

q

k

(z

2

)]

=

n

X

k=1

E

_k

h

x

2 kk

n

+

1

n

2

E

k−1

[α

∗

k

D

k

(z

1

)α

k

−

t r D

k

(z

1

)]

×

E

_k−1

[α

∗_k

D

k

(z

2

)α

k

−

t r D

k

(z

2

)]

i

=

σ

2

+

κ

n

2

n

X

k=1

X

i,j>k

[E

_k−1

D

k

(z

1

)]

i j

[E

k−1

D

k

(z

2

)]

ji

+

β

n

2

n

X

k=1

E

_k

X

i>k

[

E

_k−1

D

_k

(z

₁

)]

_ii

[

E

_k−1

D

_k

(z

₂

)]

_ii

¬

_σ

2

+

_S

₁

+

_S

₂

.

Since

E

_|

[D

_k

(z)]

_ii

₋

s(z)

_|

2

_≤

M

/(nv

4

)

uniformly on

γ

mh

,

E

_|

E

_k

[E

_k−1

D

k

(z

1

)]

ii

[E

k−1

D

k

(z

2

)]

ii

−

s(z

1

)s(z

2

)

| ≤

M

nv

4

uniformly on

γ

mh

×

γ

mh

. Hence

E

_|S

₂

₋

β

2

s(z

1

)s(z

2

)

| ≤

M

nv

4

In the following, we consider the limit of

_S

₁

. As proposed in

[

6

]

, we use the following

decomposi-tion. Let

e

_j

= (

0, . . . , 1, . . . , 0

)

′_n−1

,

j

=

1, 2, . . . ,

k

₋

1,

k

+

1, . . . ,

n

whose

j

th (or

(

j

−

1

)

th) element is 1, the rest being 0, if

j

<

k

(or

j

>

k

). Then

D

_k−1

(z) =

_p

1

n

W

n

(k)

−

z I

n−1

=

X

i,j6=k

1

p

n

x

i j

e

i

e

′

j

−

zi

n−1

z D

k

(z) +

I

=

X

i,j6=k

1

p

n

x

i j

e

i

e

′

j

D

k

(z)

We introduce the matrix

D

_{ki j}

=

₁

p

n

[W

n

(k)

−

δ

i j

(x

i j

e

i

e

′

j

+

x

ji

e

j

e

′i

)]

−

z I

n−1

−1

From the formula

A

−1

₋

B

−1

=

₋

B

−1

(A

₋

B)A

−1

, we have

D

_k

−

D

_{ki j}

=

₋

D

_{ki j}

_p

1

n

δ

i j

(

x

i j

e

i

e

′

j

+

x

ji

e

j

e

′i

)D

k

.

From the above, we get

z D

_k

(z) =

₋

I

+

X

i,j6=k

n

−1/2

x

_{i j}

e

_i

e

′_j

D

_{ki j}

(z)

−

X

i,j6=k

n

−1/2

x

_{i j}

e

_i

e

′_j

D

_{ki j}

(z)n

−1/2

δ

i j

(x

i j

e

i

e

′j

+

x

ji

e

j

e

′i

)D

k

(z)

=

₋

I

+

X

i,j6=k

n

−1/2

x

_{i j}

e

_i

e

′_j

D

_{ki j}

(z)

₋

s(z)

n

−

3

/

2

n

X

i6=k

e

_i

e

_i′

D

_k

(z)

−

X

i,j6=k

δ

i j

”

n

−1

_|

x

_{i j}

_|

2

([D

_{ki j}

(z)]

_{j j}

₋

s(z)) +

n

−1

(

_|

x

_{i j}

_|

2

₋

1

)s(z)

—

×

e

i

e

′j

D

k

(z)

−

X

i,j6=k

n

−1

δ

i j

x

2i j

[D

ki j

]

ji

e

i

e

′j

D

k

(z)

.

Therefore

z

₁

X

i,j>k

[

E

_k−1

D

_k

(z

₁

)]

_{i j}

[

E

_k−1

D

_k

(z

₂

)]

_ji

=

₋

X

i>k

[

E

_k−1

D

_k

(z

₂

)]

_ii

+n

−1/2

X

i,j,l>k

x

il

[E

k−1

D

kil

(z

1

)]

l j

[E

k−1

D

k

(z

2

)]

ji

−

s(z

₁

)

n

−

3

/

2

n

X

i,j>k

[

E

_k−1

D

_k

(z

₁

)]

_{i j}

[

E

_k−1

D

_k

(z

₂

)]

_ji

−

X

i,j>k l6=k

δ

il

E

k−1

–‚

|

x

_il

_|

2

₋

1

n

s(z

1

) +

|

x

_il

_|

2

n

[D

kil

(z

1

)

l l

−

s(z

1

)]

Œ

[D

_k

(z

₁

)]

_{i j}

™

×

[

E

_k−1

D

_k

(z

₂

)]

_ji

−

1

n

X

i,j>k l6=k

δ

il

E

k−1

x

2il

[D

kil

(z

1

)]

l i

[D

k

(z

1

)]

l j

[

E

k−1

D

k

(z

2

)]

ji

=

T

1

+

T

2

+

T

∗

+

T

3

+

T

4

.

First note that the term

T

_∗

is proportional to the term of the left hand side. We now evaluate the

contributions of the remaining four terms to the sum

_S

₁

.

The term

T

1

→

s(z

2

)

in

L

2

since

E

n

−2

X

k

X

i>k

([E

_k−1

D

_k

(z

₂

)]

_ii

₋

s(z

₂

))

2

≤

M

by

E

_|

[D

_k

(z)]

_ii

₋

s(z)

_|

2

_≤

M

nv

4

.

The terms

T

3

and

T

4

are negligible. The calculations are lengthy but simple. We omit all the details.

T

2

can not be ignored. We shall simplify it step by step. Split

T

2

as

T

2

=

X

i,j,l>k

E

_k−1

_p

x

l j

n

[D

kil

(z

1

)]

l j

[E

k−1

D

k

(z

2

)]

ji

=

X

i,j,l>k

E

_k−1

_p

x

il

n

[D

kil

(z

1

)]

l j

[E

k−1

(D

k

−

D

kil

)(z

2

)]

ji

+

X

i,j,l>k

E

_k−1

_p

x

il

n

[D

kil

(z

1

)]

l j

[

E

k−1

D

kil

(z

2

)]

ji

=

T

2a

+

T

2b

.

Again,

T

2b

can be shown to be ignored. As for the remaining term

T

2a

, we have

E

T

_2a

=

n

−2

X

i,j,l>k

x

_il

[

E

_k−1

D

_kil

(z

₁

)]

_{l j}

[

E

_k−1

(D

_k

₋

D

_kil

)(z

₂

)]

_ji

=

₋

n

−1

X

i,j,l>k

[

E

_k−1

D

_kil

(z

₁

)]

_{l j}

[

E

_k−1

D

_kil

(z

₂

)δ

il

(x

2il

e

i

e

′l

+

|

x

il

|

2

e

l

e

′i

)D

k

(z

2

)]

ji

=

J

1

+

J

2

,

where

E

_|

J

₁

_|

=

n

−1

X

i,j,l>k

E

_|

x

2_il

_|

[

E

_k−1

D

_kil

(z

₁

)]

_{l j}

[D

_kil

(z

₂

)]

_ji

[D

_k

(z

₂

)]

_{l i|}

≤

n

−1

X

i,j1,,j2,l>k

E

_|

x

_il

_|

4

_|

[

E

_k−1

D

_kil

(z

₁

)]

_{l j1}

_|

2

_|

[D

_kil

(z

₂

)]

_j2i

_|

2

×

X

i,l>k

E

_|

(D

_k

(z

₂

))

_{l i|}2

1/2

≤

M n

1/2

v

−3

.

Hence, the contribution of this term is negligible.

Finally, we have

J

2

=

−

X

i,j,l>k

E

_k−1

|

x

il

|

2

n

[D

ki j

(z

1

)]

l j

[E

k−1

D

ki j

(z

2

)]

jl

[D

k

(z

2

)]

ii

≃ −

s(z

₂

)

X

i,j,l>k

E

_k−1

1

n

[D

kil

(z

1

)]

l j

[E

k−1

D

kil

D

kil

(z

2

)]

jl

=

₋

n

−

k

n

s(z

2

)

X

j,l>k

where the last approximation follows from

X

i,j,l>k

[D

_kil

(z

₁

)]

_{l j}

[E

_k−1

D

_kil

(z

₂

)]

_jl

₋

(n

₋

k)

X

j,l>k

[E

_k−1

D

_k

(z

₁

)]

_{l j}

[E

_k−1

D

_k

(z

₂

)]

_jl

≤

M n

3/2

v

−3

.

Let us define

X

_k

=

X

i,j>k

[

E

_k−1

D

_k

(z

₁

)]

_{i j}

[

E

_k−1

D

_k

(z

₂

)]

_ji

Summarizing the estimations of all the terms

T

i

, we have proved that

z

1

X

k

=

−

s(z

1

)X

k

−

(n

−

k)s(z

2

)X

k

n

₋

k

n

+

r

k

,

where the residual term

r

k

is of order

O(

p

nv

−3

)

uniformly in

k

=

1, ...,

n

and

z

1

,

z

2

∈

γ

m

. By

z

₁

+

s(z

₁

) =

₋

1

/s(z

1

)

, the above identity is equivalent to

X

_k

= (n

₋

k)s(z

₁

)s(z

₂

) +

n

−

k

n

s(z

1

)s(z

2

)X

k

−

s(z

1

)r

k

.

Consequently,

1

n

2

n

X

k=1

X

k

=

1

n

n

X

k=1 n−k

n

s(z

1

)s(z

2

)

1

₋

n−_nk

s(z

₁

)s(z

₂

)

+

1

n

2

n

X

k=1

s(z

₁

)r

_k

1

₋

n−_nk

s(z

₁

)s(z

₂

)

,

which converges in probability to

s(z

₁

)s(z

₂

)

Z

1

0

t

1

₋

ts(z

₁

)s(z

₂

)

d t

=

−

1

−

s(z

1

)s(z

2

)

−1

log 1

−

s(z

₁

)s(z

₂

)

.

As a conclusion,

Γ

_n

(z

₁

,

z

2

)

converges in probability to

Γ(z

₁

,

z

2

) =

σ

2

−

κ

−

κ

s(z

₁

)s(z

₂

)

log 1

−

s(z

1

)s(z

2

)

₊

1

2

βs(z

1

)s(z

2

)

.

The proof of Condition 5.2 is then complete.

Although we have completed the proof of Theorem 1.1, we summarize the main steps of the proof

for reader’s convenience.

Proof of Theorem 1.1

. In Section 2, the weak convergence of

G

_n

(

f

)

is reduced to that of

G

_n

(

f

_m

)

.

Then in Section 4, we show that this is equivalent to the weak convergence of

∆ =

₋

1

2

πi

I

γm

f

_m

(z)n[s

_n

(z)

₋

s(z)]dz

.

The weak convergence of

∆

is proved in Section 5, where we also calculate the covariance function

of the limiting process. The mean function of the limiting process is obtained in Section 4.

6

Appendix

Lemma 6.1.

Under condition

(1.2)

, we have

k

E

F

_n

₋

F

_k

=

O(n

−1/2

)

,

k

F

_n

₋

F

_k

=

O

_p

(n

−2/5

)

,

k

F

n

−

F

k

=

O(n

−2/5+η

)

a.s.

,

for any

η >

0.

This follows from Theorems 1.1, 1.2 and 1.3 in

[

5

]

.

Lemma 6.2.

For X

= (x

₁

, ...,

x

_n

)

T

i.i.d. standardized (complex) entries with

E

x

_i

=

0

and

E

_|

x

_i|2

=

1

,

and C is an n

×

n matrix (complex), we have, for any p

¾

2

E

_|

X

∗

C X

₋

t r C

_|

p

¶

K

_p

”

(

E

_|

x

₁

_|

4

t r C C

∗

)

p/2

+

E

_|

x

₁

_|

2p

t r(C C

∗

)

p/2

—

.

This is Lemma 2.7 in

[

7

]

.

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