Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory
Date of Issue2017
School of Physical and Mathematical Sciences
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.
Journal of Mathematical Physics
© 2017 AIP Publishing. This paper was published in Journal of Mathematical Physics and is made available as an electronic reprint (preprint) with permission of AIP Publishing. The published version is available at: [https://doi.org/10.1063/1.5006507]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.