Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society

, Volume 29, Issue 1, pp 1–24

Central limit theorem for traces of large random symmetric matrices with independent matrix elements

Authors

  • Ya. Sinai
    • Mathematics DepartmentPrinceton University
    • Landau Institute of Theoretical Physics
  • A. Soshnikov
    • Institute for Advanced Study
Article

DOI: 10.1007/BF01245866

Cite this article as:
Sinai, Y. & Soshnikov, A. Bol. Soc. Bras. Mat (1998) 29: 1. doi:10.1007/BF01245866

Abstract

We study Wigner ensembles of symmetric random matricesA=(aij),i, j=1,...,n with matrix elementsaij,ij being independent symmetrically distributed random variables
$$a_{ij} = a_{ji} = \frac{{\xi _{ij} }}{{n^{\tfrac{1}{2}} }}.$$

We assume that Var\(\xi _{ij} = \frac{1}{4}\), fori<j, Var ξij ≤ const and that all higher moments of ξij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit theorem for the traces of powers ofA growing withn more slowly than\(\sqrt n\). The limit of Var (TraceAp),\(1 \ll p \ll \sqrt n\), does not depend on the fourth and higher moments of ξij and the rate of growth ofp, and equals to\(\frac{1}{\pi }\). As a corollary we improve the estimates on the rate of convergence of the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.

Keywords

Random matricesWigner semi-circle lawCentral limit theoremMoments

Copyright information

© Sociedade Brasileira de Matemática 1998