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=∞ 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.