Non-asymptotic Results for Singular Values of Gaussian Matrix Products

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N,  the number of terms 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 for convergence of the empirical measure of the 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=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=\infty $$\end{document} due originally to Furstenberg and Kesten (Ann Math Stat 31(2):457–469, 1960) in the 1960s, which were then further developed by Newman (Commun Math Phys 103(1):121–126, 1986) and Isopi and Newman (Commun Math Phys 143(3):591–598, 1992) as well as by a number of other authors in the 1980s. Our key technical idea is that small ball probabilities for volumes of random projections gives a way to quantify convergence in the multiplicative ergodic theorem for random matrices.