Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory

Abstract

In this paper, we consider the asymptotic behavior of X(n)fn≔∑ni=1fn(xi)Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarX(n)fn→∞VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramér type moderate deviation principle (MDP) for X(n)fnXfn(n) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramér type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case [which means fn has a particular form f(x)I(x≥θn)f(x)I(x≥θn) where θnθn is close to the right edge of equilibrium measure and f is a smooth function], X(n)fnXfn(n) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively.