Limiting behavior of large correlated Wishart matrices with chaotic entries

Abstract

We study the fluctuations, as d,n→∞, of the Wishart matrix Wn,d=1dXn,dXn,dT associated to a n×d random matrix Xn,d with non-Gaussian entries. We analyze the limiting behavior in distribution of Wn,d in two situations: when the entries of Xn,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.