Spectra of Wishart Matrices with size-dependent entries

TL;DR

This paper establishes the spectral distribution convergence of Wishart matrices with size-dependent entries, characterizing the limit law and exploring specific cases like Bernoulli and heavy-tailed variables, revealing perturbations of the Marchenko-Pastur law.

Contribution

It provides a rigorous proof of spectral measure convergence for size-dependent Wishart matrices and characterizes the limiting distribution, including explicit forms for special cases.

Findings

Limiting spectral measure converges for size-dependent Wishart matrices.

Perturbed Marchenko-Pastur distribution in Bernoulli and heavy-tailed cases.

Explicit leading term of the spectrum perturbation computed.

Abstract

We prove the convergence of the empirical spectral measure of Wishart matrices with size-dependent entries and characterize the limiting law by its moments. We apply our result to the cases where the entries are Bernoulli variables with parameter c=n or truncated heavy-tailed random variables. In both cases, when c goes to infinity or when the truncation is small, the limiting spectrum is a perturbation of the Marchenko-Pastur distribution and we compute its leading term.