High-dimensional regime for Wishart matrices based on the increments of the solution to the stochastic heat equation

Abstract

We consider a $\mathit{n}\times \mathit{d}$ random matrix ${\mathcal{X}}_{\mathit{n},\mathit{d}}$ whose entries are the spatial increments of the solution to the stochastic heat equation with space-time white noise. We analyze the limit behavior of the associated Wishart matrix, by showing that it converges almost surely to a diagonal matrix (with equal diagonal terms) and the renormalized Wishart matrix satisfies a central limit theorem. Our techniques are based on the analysis on Wiener chaos, Malliavin calculus and Stein’s method.