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

Girko, V. L.

Teor. Veroyatn. Mat. Stat., Kiev 41, 23-29 (1989).

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

Keywords:

asymptotics of the eigenvalues distribution; semicircle law; empirical covariance matrix

Citations:

Zbl 0268.60034