## Limit theorems for the distribution of eigenvalues of random symmetric matrices.(Russian)Zbl 0698.60049

The paper contains the proof of the following
Theorem 1. Let $$A+\Xi$$ be a real symmetric matrix where $$A=\{a_ i\delta_{ij}\}$$, $$i,j=1,...,n$$ is diagonal and $$\Xi =\{\xi_{ij}^{(n)}\}$$ is random. Suppose that $$| a_ i| \leq c<\infty$$ and $$\xi_{ij}^{(n)}$$ are independent for $$i\geq j$$ and identically distributed random variables, $E \xi_{ij}^{(n)}=0,\quad E [\xi_{ij}^{(n)}]^ 2=n^{-1},\quad E \{[\xi_{ij}^{(n)}n^{1/2}]^ 2-1\}^ 2-2=\alpha_{ij}$ and for some $$\beta >0$$ $\sup_{n}\sup_{i,j}E | \xi_{ij}^{(n)}n^{1/2}|^{4+\beta}<\infty.$ If $$\{\nu_ i\}$$, $$i=1,...,n$$ are eigenvalues of $$A+\Xi$$, then for $$\epsilon_ n^{-1}=o(n^{\gamma /q})$$ with $$\gamma >0$$ and integer $$q>0,$$ $$(\partial /\partial x)\sum^{n}_{k=1}P\{\nu_ k+\epsilon_ k\eta <x\}=\pi^{-1}Im\{nc(z)+f(z)+o(1)\},\quad where$$ $f(z)=(1-\partial c/\partial z)\{n^{-1}\sum^{n}_{k=1}c^ 2_ k(\partial /\partial z)\ell n c_ k+n^{-1}\sum^{n}_{k=1}c^ 3_ k[1+2(\partial c/\partial z)+n^{-1}\sum^{n}_{\ell =1}c^ 2_{\ell}\alpha_{\ell k}]\}.$ Here $$\eta$$ is a Cauchy distributed random variable and independent of $$\Xi$$, c(z) is an analytic function, if $$\epsilon_ n\neq 0$$, satisfying the equation $c(z)=n^{-1}\sum^{n}_{1}c_ k(z),\quad c_ k(z)=[z-q_ k-c(z)]^{-1},\quad z=x+i\epsilon_ n.$ Thus, the author found the remainder term in the well-known Wigner semicircle law [see e.g. the reviewer, Uspehi mat. Nauk 28, No.1(169), 3- 64 (1973; Zbl 0268.60034)]. A similar theorem is also proved for the empirical covariance matrix of $$n+1$$ independent identically distributed $$m_ n$$-dimensional Gaussian vectors if $$\overline{\lim_{}n\to \infty}m_ nn^{-1}<\infty$$.
Reviewer: L.Pastur

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 15B52 Random matrices (algebraic aspects) 60F99 Limit theorems in probability theory

Zbl 0268.60034