NON-ASYMPTOTIC RESULTS FOR SINGULAR VALUES OF GAUSSIAN MATRIX PRODUCTS
NON-ASYMPTOTIC RESULTS FOR SINGULAR VALUES OF GAUSSIAN MATRIX PRODUCTS
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DOI：
10.1007/s00039-021-00560-w
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发表时间：
2021-05-08
影响因子：
2.2
通讯作者：
Paouris, Grigoris
中科院分区：
文献类型：
--
作者：
Hanin, Boris;Paouris, Grigoris
本文提供了一个非渐近分析的奇异值（和李雅普诺夫指数）的高斯矩阵产品的制度，其中N，在产品中的项数，是大的和n，矩阵的大小，可以是大的或小的，并可能取决于N。我们得到了李雅普诺夫指数和的浓度估计，奇异值平方的经验测度收敛到[0，1]上的均匀分布的定量速度，以及当N作为n的函数充分大时李雅普诺夫指数的联合正态性结果.我们的技术包括N=无穷大时遍历理论方法的非渐近版本，最初由Furstenberg和Kesten（Ann Math Stat 31（2）：457-469，1960）在20世纪60年代提出，然后由纽曼（Commun Math Phys 103（1）：121-126，1986）和Isopi和纽曼（Commun Math Phys 143（3）：591-598，1992）以及其他一些作者在20世纪80年代进一步发展。我们的关键技术思想是，小球概率体积的随机投影给出了一种方法来量化的随机矩阵的乘法遍历定理的收敛。
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=infinity 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.