Limiting behavior of large correlated Wishart matrices with chaotic entries

TL;DR

This paper investigates the asymptotic distribution of large Wishart matrices with chaotic entries, showing Gaussian limits for independent entries and non-Gaussian limits for correlated entries, using advanced probabilistic techniques.

Contribution

It provides a detailed analysis of the limiting behavior of Wishart matrices with non-Gaussian, chaotic entries, extending understanding beyond classical Gaussian cases.

Findings

Gaussian limit for independent chaotic entries

Non-Gaussian diagonal limit for correlated entries

Explicit convergence rates in Wasserstein distance

Abstract

We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix $\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T} $ associated to a $n\times d$ random matrix $\mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $\mathcal{W}_{n,d}$ in two situations: when the entries of $\mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.