 
Summary: A CLT FOR A BAND MATRIX MODEL
GREG ANDERSON AND OFER ZEITOUNI
Abstract. A law of large numbers and a central limit theorem are derived
for linear statistics of random symmetric matrices whose onorabove diago
nal entries are independent, but neither necessarily identically distributed, nor
necessarily all of the same variance. The derivation is based on systematic
combinatorial enumeration, study of generating functions, and concentration
inequalities of the Poincar’e type. Special cases treated, with an explicit eval
uation of limiting variances, are generalized Wigner and Wishart matrices.
1. Introduction
The interest in the limiting properties of the empirical distribution of eigenvalues
of large symmetric random matrices can be traced back to [Wis28] and to the path
breaking article of Wigner [Wig55]. We refer to [Ba99], [De00], [HP00], [Me91] and
[PL03] for partial overview and some of the recent spectacular progress in this field.
In this paper we study both convergence of the empirical distribution and central
limit theorems for linear statistics of the empirical distribution of a class of random
matrices. To give right away the flavor of our results, consider for each positive
integer N the NbyN symmetric random matrix X(N) with onorabovediagonal
entries X(N) ij = N 1/2 f(i/N, j/N) 1/2 # ij , where the # ij are zero mean unit vari
ance i.i.d. random variables satisfying the Poincar’e inequality with constant c (see
