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 \(N=\infty \) 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.

本文提供了一种高斯矩阵乘积的奇异值（和Lyapunov指数）的非渐近分析，其中N是乘积项的个数很大，而n是矩阵的大小，或者小并且可以取决于ñ。我们获得李雅普诺夫指数总和的浓度估计值，奇异值平方的实测值收敛于[0，1]上均匀分布的定量速率，以及当N足够大时李雅普诺夫指数的联合正态性的结果。n的函数。我们的技术由遍历理论方法的非渐近形式组成，位于\（N = \ infty \）最初是由Furstenberg和Kesten（Ann Math Stat 31（2）：457–469，1960）于1960年代提出的，后来由Newman（Commun Math Phys 103（1）：121–126，1986）和Isopi和纽曼（Commun Math Phys 143（3）：591–598，1992），以及1980年代的许多其他作者的著作。我们的主要技术思想是，针对随机投影量的小球概率为量化随机矩阵的遍历遍历定理中的收敛提供了一种方法。