Non-asymptotic Results for Singular Values of Gaussian Matrix Products

Abstract: This article concerns the non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where $N,$ the number of term in the product, is large and $n,$ the size of the matrices, may be large or small and may depend on $N$. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate of convergence of the empirical measure of the normalized squared singular values to the uniform distribution on $[0,1]$, and results on the joint normality of Lyapunov exponents when $N$ is sufficiently large as a function of $n.$ Our technique consists of non-asymptotic versions of the ergodic theory approach at $N=\infty$ due originally to Furstenberg and Kesten in the 1960's, which were then further developed by Newman and Isopi-Newman as well as by a number of other authors in the 1980's. Our key technical idea is that small ball probabilities for volumes of random projections give a way to quantify convergence in the multiplicative ergodic theorem for random matrices.