论文标题
高斯基质产物奇异值的非反应结果
Non-asymptotic Results for Singular Values of Gaussian Matrix Products
论文作者
论文摘要
本文涉及对制度中高斯矩阵产品的奇异值（和Lyapunov指数）的非反应分析，在该制度中，$ n，$ n，$ n of the Product的术语数量很大，$ n，$ n，$ the矩阵的大小，可能是大小的，可能是大的，可能是$ n $的。我们获得了Lyapunov指数总和的浓度估计值，这是对归一化平方的经验度量的定量收敛速率，即$ [0,1] $上的均匀分布的经验度量，以及$ n $的共同正态性的结果，当$ n $的联合正态性与我们的技术相关的方法很大。 $ n = \ infty $最初是由于1960年代的Furstenberg和Kesten造成的，然后由Newman和Isopi-Newman以及1980年代的许多其他作者进一步开发。我们的关键技术思想是，随机投影的小球概率为量化乘法的ergodic定理中的收敛性提供了一种方法。
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.
