On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-Identically Distributed Entries

TL;DR

This paper extends fluctuation results of matrix entries for regular functions of Wigner matrices to non-i.i.d. cases, under specific moment conditions and relaxed test function regularity assumptions.

Contribution

It generalizes previous results to non-i.i.d. Wigner matrices with moment conditions and less restrictive test function requirements.

Findings

Fluctuations of matrix entries are characterized under new conditions.

Results apply to Wigner matrices with non-identically distributed entries.

Relaxed regularity conditions on test functions are established.

Abstract

In this note, we extend the results about the fluctuations of the matrix entries of regular functions of Wigner random matrices obtained in arXiv:1103.3731 [math.PR] to Wigner matrices with non-i.i.d. entries provided certain Lindeberg type conditions for the fourth moments of the off-diagonal entries and the second moments of the diagonal entries are satisfied. In addition, we relax our conditions on the test functions and require that for some $s>3 \ \int (1+|k|)^{2s}\*|\hat{f}(k)|^2 \* dk <\infty.$