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

and Applied Analysis 3


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.An  ×  Wigner matrix is defined to be a random Hermitian matrix   = (  ) 1≤,≤ , in which the real and imaginary parts of   for  <  are i.i.d.random variables with mean 0 and variance 1/2 (in that case, E 2  = 0), and   ,  = 1, 2, . . ., , are i.i.d.random variables with mean 0 and variance 1. Denote by  1 ≤ ⋅ ⋅ ⋅ ≤   the real eigenvalues of the normalized Wigner matrix   = (1/√)  .The classical Wigner theorem states that the empirical distribution    () = (1/) ∑  =1 1 {  ≤} converges almost surely to the semicircle law with the density   () = (1/2) √ 4 −  2  (1) 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 matrix   satisfy that |  | 4 = 2 and |  | 6 < ∞ for all 1 ≤ ,  ≤ .Let Π be an open interval including the interval [−2, 2] and  4 (Π) is the space of forth-order continuous differentiable functions on Π. Denote   () :=  {∫  ()    () − ∫  ()   () } .(2) Then the empirical process {  () :  ∈  4 (Π)} converges weakly in finite dimension to a Gaussian process {() :  ∈  4 (Π)} with mean zero and the covariance function Cov (, ) given by Cov (, ) )  . ( In addition, Guionnet and Zeitouni [2] give a concentration inequality on the empirical spectral measure of    () near the semicircle law.Lemma 2. Assume the law of the entries {Re(  ), Im(  ),   } satisfies the logarithmic Sobolev inequality (LSI) with constant  0 > 0. Let  be a Lipschitz function and denote Then for any  >   () = |E ∫ ()   () − ∫ ()  ()|, there exists  > 0, such that 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.
where  ∼ (0, 1), and We will give the proofs of theorems in Section 2. 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 ,  1 ,  2 , . .., be i.i.d.random variables and   = ∑  =1   ; there are a number of results on the convergence of a type of series where () and () are the positive functions defined on [0, ∞), and ∑ ∞ =1 () = ∞.In fact, the sum (11) tends to infinity when  ↘ 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 () such that the sum (11) multiplied by () 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 lim ↘0  2 ∑ ∞ =1 (|  | ≥ ) = E 2 under the assumptions that E = 0 and E 2 < ∞.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 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 lim when E = 0, E 2 =  2 , and E 2 log + || < ∞.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 write (, ) = [ −1/ ], where  > 0 and  is an arbitrary positive real number.‖‖ = sup ∈ |()|, and  stands for a random variable observing the standard Gaussian distribution.We also denote  to be the absolutely positive constant whose value can be different from one place to another.

The Proofs
Before the main proof of Theorem 3, we will first give four propositions.

Proposition 5. Under the assumptions of Theorem 3, one has
Proof.We calculate that lim
Proof of Theorem 3.According to the fact that for any random variable  and  ∈ R, we can see that By Propositions 5-8, we can get Theorem 3 easily.
Proof of Theorem 4. The proof is essentially the same as for Theorem 3, 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]).Just like the proof of Proposition 6, in order to use Lemma 10, we will need to estimate the order of δ, (), and the following remark is needed.