# Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

Authors: Junshan Xie
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper considers the precise asymptotics of the spectral statistics of random matrices. Following the ideas of Gut and Spătaru (2000) and Liu and Lin (2006) on the precise asymptotics of i.i.d. random variables in the context of the complete convergence and the second-order moment convergence, respectively, we will establish the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of Wigner matrices or sample covariance matrices.


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## Body

## 1. Introduction and Main Results

This paper is concerned with the precise asymptotic behaviors of the spectral (eigenvalue) statistics of random matrices; two types of classical random matrices including Wigner matrices and sample covariance matrices will be considered. Ann×n Wigner matrix is defined to be a random Hermitian matrix Xn=(Xij)1≤i,j≤n, in which the real and imaginary parts of Xij for i<j are i.i.d. random variables with mean 0 and variance 1/2 (in that case, 𝔼Xij2=0), and Xii, i=1,2,…,n, are i.i.d. random variables with mean 0 and variance 1. Denote by λ1≤⋯≤λn the real eigenvalues of the normalized Wigner matrix Mn=(1/n)Xn. The classical Wigner theorem states that the empirical distribution FMn(x)=(1/n)∑i=1n‍1{λi≤x} converges almost surely to the semicircle law with the density μsc(x)=(1/2π)4-x21[-2,2](x). Consequently, for any bounded continuous function f, the spectral statistics satisfy that
(1)limn→∞1n∑i=1n‍f(λi)=limn→∞∫‍f(x)dFMn(x)=∫‍f(x)μsc(x)dxa.s.The result can be viewed as an analog of the law of large number for independent random variables. As for the fluctuation of the spectral statistics, a remarkable work due to Bai et al. [1] states the following.Lemma 1.
Assume the entries of a Wigner matrixXn satisfy that E|Xij|4=2 and E|Xij|6<∞ for all 1≤i, j≤n. Let Π be an open interval including the interval [-2,2] and C4(Π) is the space of forth-order continuous differentiable functions on Π. Denote
(2)Gn(f)≔n{∫‍f(x)dFMn(x)-∫‍f(x)μsc(x)dx}.
Then the empirical process {Gn(f):f∈C4(Π)} converges weakly in finite dimension to a Gaussian process {G(f):f∈C4(Π)} with mean zero and the covariance function Cov(f,g) given by
(3)Cov(f,g)=14π2∬-22f′(t)g′(s)=14π2×log(4-ts+(4-t2)(4-s2)4-ts-(4-t2)(4-s2))dtds.In addition, Guionnet and Zeitouni [2] give a concentration inequality on the empirical spectral measure of FMn(x) near the semicircle law.Lemma 2.
Assume the law of the entries{Re(Xij),Im(Xij),Xii} satisfies the logarithmic Sobolev inequality (LSI) with constant c0>0. Let f be a Lipschitz function and denote
(4)ηn(f)=n-1Gn(f)=∫‍f(x)dFMn(x)-∫‍f(x)μsc(x)dx.
Then for any δ>δn(f)=|𝔼∫‍f(x)dFMn(x)-∫‍f(x)μsc(x)dx|, there exists c>0, such that
(5)P(|ηn(f)|≥δ)≤2e-cn2(δ-δn(f))2.Based on the above existing results on the limiting spectral properties of random matrices, we will give another type of asymptotic spectral properties, that is, the precise asymptotics of the spectral statistics of random matrices. In particular, we will consider the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of random matrices. Our first result can be listed as follows.Theorem 3.
Suppose that the law of the entries{Re(Xij),Im(Xij),Xii} satisfies LSI with constant c0>0 and E|Xij|4=2 for all 1≤i, j≤n. Denote ηn(f) as (4), where f∈C4(Π) has a bounded first-order derivative. Then for any s>1/2, β>2s-1, one has
(6)limϵ↘0ϵ(β+1)/s-2∑n=1∞‍nβ-2s+2𝔼|ηn(f)|2I{|ηn(f)|≥ϵns-1}=1β-2s+1[D(f)](β+1)/2s𝔼|ξ|(β+1)/s,
where ξ is a standard Gaussian random variable and
(7)D(f)=14π2∬-22f′(t)f′(s)×log(4-ts+(4-t2)(4-s2)4-ts-(4-t2)(4-s2))dtds.Now we turn to the sample covariance matrix, which is an important statistic in multivariate statistics analysis. LetYmn=(Yij)1≤i≤m,1≤j≤n be an m×n random matrix whose entries are i.i.d. complex-valued random variables with mean 0 and variance 1 and Re(Yij)i<j and Im(Yij)i<j are independent random variables with mean 0 and variance 1/2. Then Wm,n=(1/n)YmnYmn* can be viewed as the sample covariance matrix of n samples of m-dimensional random vectors. Let λ1′≤⋯≤λm′ be the eigenvalues of Wm,n, and define the empirical spectral distribution FWm,n(x)=(1/m)∑i=1m‍I{λi′≤x}. When m/n→ρ∈(0,1] as n→∞, the famous Marchenko-Pastur theorem reveals that almost surely FWm,n(x) converges to the Marchenko-Pastur law with the density γρ(x)=(1/2πxρ)(b(ρ)-x)(x-a(ρ))1[a(ρ),b(ρ)](x), where a(ρ)=(1-ρ)2, b(ρ)=(1+ρ)2.Denote(8)τm,n(f)=∫‍f(x)dFWm,n(x)-∫‍f(x)γm/n(x)dx,
where γm/n(x) is the Marchenko-Pastur law γρ(x) with the parameter ρ=m/n. Similar to the result on Wigner matrices, we can give the result on the complex sample covariance matrices.Theorem 4.
Assume the law of the entries{Re(Yij),Im(Yij)} satisfies LSI with constant c0>0 and E|Yij|4=2 for all 1≤i≤m, 1≤j≤n, m=[nρ](0<ρ≤1). If one denotes Γ to be an open interval including [a(ρ),b(ρ)], f∈C4(Γ) has a bounded first order derivative. Then for any s>1/2, β>2s-1, one has
(9)limϵ↘0ϵ(β+1)/s-2∑n=1∞‍nβ-2s+2𝔼|τm,n(f)|2I{|τm,n(f)|≥ϵns-1}=1β-2s+1ρ(β+1)/s[V(f)](β+1)/2s𝔼|ξ|(β+1)/s,
where ξ~N(0,1), and
(10)V(f)=12π2∬a(ρ)b(ρ)f′(x1)f′(x2)×log|s_(x1)¯-s_(x2)s_(x1)-s_(x2)|dx1dx2,s_(x)=limz→x+i0s_(z) and s_(z) is the Stieltjes transform of F_2(y)=(1-y)1[0,∞)+y∫-∞y‍γρ(x)dx, that is, s_(z)=∫-∞+∞‍(1/(x-z))dF_2(x).We will give the proofs of theorems in Section2. Below are a few words about the motivation of this paper. In a sense, our results are similar to the precise asymptotics of independent random variables in the context of complete convergence and moment convergence. Let X,X1,X2,…, be i.i.d. random variables and Sn=∑i=1n‍Xi; there are a number of results on the convergence of a type of series
(11)∑n=1∞‍φ(n)P(|Sn|≥ϵf(n)),ϵ>0,
where φ(x) and f(x) are the positive functions defined on [0,∞), and ∑n=1∞‍φ(n)=∞. In fact, the sum (11) tends to infinity when ϵ↘0, one of the interesting problems is to examine the precise rate at which this occurs, and this amounts to finding a suitable normalizing rate function ψ(ϵ) such that the sum (11) multiplied by ψ(ϵ) has a nontrivial limit; this kind of results is frequently called “precise asymptotics.” The first result in this direction due to Heyde [3], who proved that limϵ↘0ϵ2∑n=1∞P(|Sn|≥ϵn)=𝔼X2 under the assumptions that 𝔼X=0 and 𝔼X2<∞. Some analogous results in more general case can be found in Gut and Spǎtaru [4, 5] and Gut and Steinebach [6]. Moreover, Chow [7] studied the convergence properties of the series ∑n=1∞npα-2-α𝔼{max1≤j≤n|Sj|-ϵnα}+(ϵ>0) when p≥1, α>1/2, pα>1. There is a remarkable result obtained by Liu and Lin [8], who considered the precise asymptotics on the second-order moment convergence, which states that
(12)limϵ↘01-logϵ∑n=1∞‍n-2𝔼Sn2I{|Sn|≥ϵn}=2σ2
when 𝔼X=0, 𝔼X2=σ2, and 𝔼X2log+|X|<∞. Chen and Zhang [9] got the similar results on the second-order moment convergence of empirical process. Furthermore, there are also some precise asymptotic results in other contexts, such as the self-normalized sums, martingale-difference, random fields, and renewal process. It should be mentioned that the corresponding results on random matrices and random growth model have been studied by Su [10], who presented the precise asymptotics of the largest eigenvalues of Gaussian unitary ensembles, Laguerre unitary ensembles, and the longest increasing subsequence of a random permutation. In this paper, we will study the precise asymptotics on the spectral statistics of Wigner matrices and sample covariance matrices, which is also an interesting topic in random matrix theory.In the rest of the paper, we will always writeN(ϵ,M)=[Mϵ-1/s], where ϵ>0 and M is an arbitrary positive real number. ∥f∥=supx∈R|f(x)|, and ξ stands for a random variable observing the standard Gaussian distribution. We also denote C to be the absolutely positive constant whose value can be different from one place to another.


## 2. The Proofs

Before the main proof of Theorem3, we will first give four propositions.Proposition 5.
Under the assumptions of Theorem3, one has
(13)limϵ↘0ϵ(β+1)/s∑n=1∞‍nβP(|ξ|≥ϵnsD(f))=1β+1[D(f)](β+1)/2sE|ξ|(β+1)/s.Proof.
We calculate that(14)limϵ↘0ϵ(β+1)/s∑n=1∞‍nβP(|ξ|≥ϵnsD(f))=limϵ↘0ϵ(β+1)/s∫1∞‍yβP(|ξ|≥ϵysD(f))dy=1s[D(f)](β+1)/2s∫0∞‍t(β+1)/s-1P(|ξ|≥t)dt=1β+1[D(f)](β+1)/2sE|ξ|(β+1)/s.Proposition 6.
Under the assumptions of Theorem3, one has
(15)limϵ↘0ϵ(β+1)/s∑n=1∞‍nβ|P(|nηn(f)|≥ϵns)-P(|ξ|≥ϵnsD(f))|=0.Proof.
We can write(16)∑n=1∞‍nβ|P(|nηn(f)|≥ϵns)-P(|ξ|≥ϵnsD(f))|≤∑n≤N(ϵ,M)‍nβ|P(|nηn(f)|≥ϵns)-P(|ξ|≥ϵnsD(f))|+∑n>N(ϵ,M)‍nβP(|nηn(f)|≥ϵns)+∑n>N(ϵ,M)‍nβP(|ξ|≥ϵnsD(f))≔I1+I2+I3.
Under the assumptions of Theorem3, the law of each entry Xij also satisify LSI. By Lemma 5 of Vershynin [11], for each fixed positive integer k, 𝔼|Xij|k<+∞. According to Lemma 1, we can see that
(17)nηn(f)D(f)⟶dN(0,1).
Thus, as n→∞,
(18)Δn=:supx∈ℝ|P(|nηn(f)|≥x)-P(|ξ|≥xD(f))|⟶0.
Using Toeplitz's lemma, we can deduce that
(19)limϵ↘0ϵ(β+1)/sI1=0.
For the termI2, if f∈C4(Π) has a bounded first-order derivative, then f is a Lipschitz function.
By Theorem 8.2 of Bai and Silverstein [12], there exists a real number C1>0, such that
(20)δn(f)=|𝔼∫‍f(x)dFMn(x)-∫‍f(x)μsc(x)dx|≤C1∥f′∥n-1/2.
By Lemma 2, for any δ0>nδn(f), there exists c>0 such that
(21)P(|nηn(f)D(f)|≥δ0)≤2e-c(δ0-nδn(f))2.
For each fixedϵ>0, the assumption s>1/2 reveals that, when n>N(ϵ,M) with a sufficient large number M, we have ϵns>nδn(f). Thus there exists c1>0, such that
(22)limM→∞I2≤limM→∞C∫N(ϵ,M)∞‍xβe-c1x2sdx=0.
For the termI3, as the fact that ξ~N(0,1), the well-known tail probability estimation states that for any x∈ℝ,
(23)P(ξ≥x)≤12πxe-x2/2.
Hence, for any fixed ϵ>0, we can get
(24)limM→∞I3≤limM→∞C∫N(ϵ,M)∞‍xβ-1e-x2/2dx=0.
By letting ϵ↘0 and then M→∞, a combination of (22) and (24) can get that limϵ↘0ϵ(β+1)/s(I2+I3)=0. Then combined with the relations (16) and (19), we can complete the proof.Proposition 7.
Under the assumptions of Theorem3, one has
(25)limϵ↘0ϵ(β+1)/s-2∑n=1∞‍nβ-2s∫ϵns∞‍yP(|ξ|≥yD(f))dy=s(β+1)(β-2s+1)[D(f)](β+1)/2s𝔼|ξ|(β+1)/s.Proof.
By the similar argument in the proof of Proposition5, we can show that
(26)limϵ↘0ϵ(β+1)/s-2∑n=1∞‍nβ-2s∫ϵns∞‍yP(|ξ|≥yD(f))dy=limϵ↘0ϵ(β+1)/s-2D(f)∫1∞‍xβ-2s∫ϵxs/D(f)∞‍yP(|ξ|≥y)dy=limϵ↘02s[D(f)]1/2s∫ϵ/D(f)∞‍t1/s-3∫t∞‍yP(|ξ|≥y)dy=2s[D(f)](β+1)/2s∫0∞‍yP(|ξ|≥y)∫0y‍t(β+1)/s-3dtdy=s(β+1)(β-2s+1)[D(f)](β+1)/2s𝔼|ξ|(β+1)/s.Proposition 8.
Under the assumptions of Theorem3, one has
(27)limϵ↘0ϵ(β+1)/s-2∑n=1∞‍nβ-2s|(|ξ|≥yD(f))∫ϵns∞‍yP(|nηn(f)|≥y)dy-∫ϵns∞‍yP(|ξ|≥yD(f))dy|=0.Proof.
Similar to the argument in the proof of Proposition6, we can write
(28)∑n=1∞‍nβ-2s|(|ξ|≥yD(f))∫ϵns∞‍yP(|nηn(f)|≥y)dy-∫ϵns∞‍yP(|ξ|≥yD(f))dy|=∑n=1N(ϵ,M)‍nβ-2s|(|ξ|≥yD(f))∫ϵns∞‍yP(|nηn(f)|≥y)dy-∫ϵns∞‍yP(|ξ|≥yD(f))dy|+∑n>N(ϵ,M)‍nβ-2s|(|ξ|≥yD(f))∫ϵns∞‍yP(|nηn(f)|≥y)dy-∫ϵns∞‍yP(|ξ|≥yD(f))dy|≔J1+J2.
For the termJ1, we have
(29)J1≤∑n=1N(ϵ,M)‍nβ-2s×∫ϵns∞‍y|P(|nηn(f)|≥y)-P(|ξ|≥yD(f))|dy≤∑n=1N(ϵ,M)‍nβ∫0n-sΔn-1/4‍(x+ϵ)×|P(|nηn(f)|≥(x+ϵ)ns)-P(|ξ|≥(x+ϵ)nsD(f))|dx+∑n=1N(ϵ,M)‍nβ×∫n-sΔn-1/4∞‍(x+ϵ)P(|nηn(f)|≥(x+ϵ)ns)dx+∑n=1N(ϵ,M)‍nβ×∫n-sΔn-1/4∞‍(x+ϵ)P(|ξ|≥(x+ϵ)nsD(f))dx≔∑n=1N(ϵ,M)‍nβ(J11+J12+J13).
As N(ϵ,M)=[Mϵ-1/s], for n≤N(ϵ,M), we have ϵ≤Msn-s. Thus
(30)n2sJ11≤n2s∫0n-sΔn-1/4‍(x+ϵ)Δndx≤Cn2sΔn(n-sΔn-1/4+Msn-s)2=C(Δn1/4+MsΔn1/2)2⟶0asn⟶∞.
Noticing that whenϵ>0 is fixed, ϵns>nδn(f) for sufficient large n, the relation (21) tells us that there exists c1>0, such that
(31)limn→∞n2sJ12=limn→∞n2s∫n-sΔn-1/4∞‍(x+ϵ)P(|nηn(f)|≥(x+ϵ)ns)dx=limn→∞∫ϵns+Δn-1/4∞‍yP(|nηn(f)|≥y)dy≤limn→∞C∫ϵns+Δn-1/4∞‍ye-c1y2dy≤limn→∞C∫Δn-1/4∞‍ye-c1y2dy≤limn→∞Ce-c1Δn-1/2=0.
By the same argument as forJ12, using the relation (23), we can easily prove that
(32)limn→∞n2sJ13=0.
Hence, we have proved thatlimn→∞Ξn(ϵ)≔limn→∞n2s(J11+J12+J13)=0 uniformly for ϵ>0. By Toeplitz's lemma again, we can further deduce that
(33)limϵ↘0ϵ(β+1)/s-2J1=limϵ↘0ϵ(β+1)/s-2∑n=1N(ϵ,M)‍nβ(J11+J12+J13)≤limϵ↘0C[N(ϵ,M)]2s-β-1∑n=1N(ϵ,M)‍nβ-2sΞn(ϵ)=0.
For the termJ2, we can write
(34)J2≤∑n>N(ϵ,M)‍nβ-2s×∫ϵns∞‍y[P(|nηn(f)|≥y)+P(|ξ|≥yD(f))]dy≔J21+J22.
By using the relation (21) again, for any fixed ϵ>0, when M→∞, there exists c1>0, such that
(35)ϵ(β+1)/s-2J21≤Cϵ(β+1)/s-2∑n>N(ϵ,M)‍nβ-2s∫n∞‍ϵ2x2s-1P(|nηn(f)|≥ϵxs)dx≤Cϵ(β+1)/s∑n>N(ϵ,M)‍nβ-2s∫n∞‍x2s-1e-c1ϵ2x2sdx≤Cϵ(β+1)/s∫N(ϵ,M)∞‍yβ-2s∫y∞‍x2s-1e-c1ϵ2x2sdxdy≤Cϵ(β+1)/s∫N(ϵ,M)∞‍xβ-2s+1x2s-1e-c1ϵ2x2sdx≤C∫Ms∞‍y(β+1)/s-1e-c1y2dy⟶0.
Hence, by letting ϵ↘0 firstly and then taking M→∞, we get limϵ↘0ϵ(β+1)/s-2J21=0.
Following the proof ofJ21 and using the relation (23), we can easily prove that ϵ(β+1)/s-2J22 vanishes as ϵ↘0, which yields that
(36)limϵ↘0ϵ(β+1)/s-2J2=0.
By combining (33) and (36), we conclude that the proof is completed.Proof of Theorem3.
According to the fact that for any random variableζ and a∈ℝ,
(37)𝔼ζI{ζ≥a}=aP(ζ≥a)+∫a∞‍P(ζ≥t)dt,
we can see that
(38)𝔼|ηn(f)|2I{|ηn(f)|≥ϵns-1}=ϵ2n2s-2P(|ηn(f)|≥ϵns-1)+2n-2∫ϵns∞‍yP(|nηn(f)|≥y)dy.
By Propositions5–8, we can get Theorem 3 easily.Proof of Theorem4.
The proof is essentially the same as for Theorem3, which mainly depends on the central limit theorem of the spectral statistics and the concentration inequality of the empirical spectral measure. We will only list some key tools in the proof, and the details are omitted here.Lemma 9 (see [13]).
Consider the sample covariance matrixWm,n=(1/n)YmnYmn* as above, where E|Yij|4=2 and E|Yij|8<∞ for all 1≤i≤m, 1≤j≤n, m=[nρ](0<ρ≤1). Let
(39)G~m,n(f)=m{∫‍f(x)dFWm,n(x)-∫‍f(x)γm/n(x)dx}
and denote Γ to be an open interval including [a(ρ),b(ρ)]. Then the empirical processes {G~m,n(f):f∈C4(Γ)} converge weakly in finite dimension to a Gaussian process {G~(f):f∈C4(Γ)} with mean zero and the covariance function Cov~(f,g) given by
(40)Cov~(f,g)=12π2∬a(ρ)b(ρ)f′(x1)g′(x2)×log|s_(x1)¯-s_(x2)s_(x1)-s_(x2)|dx1dx2,f,g∈C4(Γ),
where s_(x) is defined in Theorem 4.Lemma 10.
Assumef is a Lipschitz function and τm,n(f) is defined by (8). Denote δ~m,n(f)=|𝔼∫‍f(x)dFWm,n(x)-∫‍f(x)γm/n(x)dx|. Then under the assumptions of Theorem 4, for any δ>δ~m,n(f), there exists c2>0, such that
(41)P(|τm,n(f)|≥δ)≤2e-c2m2(δ-δ~m,n(f))2.Proof.
If we denote(42)τm,n∘(f)=∫‍f(x)dFWm,n(x)-E[∫‍f(x)dFWm,n(x)],
then by the Corollary 1.8(b) in Guionnet and Zeitouni [2], for any δ>0, there exists c2>0 such that
(43)P(|τm,n∘(f)|≥δ)≤2e-c2δ2m2.
As we can write(44)P(|τm,n(f)|≥δ)≤P(|𝔼∫‍f(x)dFWm,n(x)-∫‍f(x)γm/n(x)dx||τm,n∘(f)|+|𝔼∫‍f(x)dFWm,n(x)-∫‍f(x)γm/n(x)dx|≥δ)≤P(|τm,n∘(f)|≥δ-δ~m,n(f)),
for each δ>δ~m,n(f), we have
(45)P(|τm,n(f)|≥δ)≤2e-c2m2(δ-δ~m,n(f))2.Just like the proof of Proposition6, in order to use Lemma 10, we will need to estimate the order of δ~m,n(f), and the following remark is needed.Remark 11.
For the estimation ofδ~m,n(f), by the result of Bai et al. [14], we can see that
(46)supx∈ℝ|𝔼F2(m,n)(x)-F2(m,n)(x)|≤Cm-1/2,
where F2(m,n)(x)=∫0x‍dFWm,n(t) is the empirical spectral distribution of Wm,n. As a direct consequence, when f is differentiable and f′ is bounded, there exists a real number C2>0, such that δ~m,n(f)≤C2∥f′∥m-1/2.



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