A non-Gaussian limit for linear eigenvalue statistics of Hankel matrices

TL;DR

This paper demonstrates that for Hankel matrices with independent entries, the linear eigenvalue statistics for odd degree monomials greater than or equal to three do not follow a Gaussian distribution, contrasting with even degree cases.

Contribution

It establishes a non-Gaussian limit for certain linear eigenvalue statistics of Hankel matrices, extending understanding beyond previously known Gaussian cases.

Findings

Odd degree monomials ≥ 3 do not converge to Gaussian distribution.

Contrasts with Gaussian limits for even degree monomials.

Uses convergence of moments to prove non-Gaussian behavior.

Abstract

This article focuses on linear eigenvalue statistics of Hankel matrices with independent entries. Using the convergence of moments we show that the linear eigenvalue statistics of Hankel matrices for odd degree monomials with degree greater than or equal to three does not converge in distribution to a Gaussian random variable. This result is a departure from the known results, Liu, Sun and Wang (2012), Kumar and Maurya (2022), of linear eigenvalue statistics of Hankel matrices for even degree monomial test functions, where the limits were Gaussian random variables.