Large-deviation asymptotics of condition numbers of random matrices. (December 2021)

• Record Type: Journal Article
• Authors: Singull, Martin; Uwamariya, Denise; Yang, Xiangfeng
• Is Part Of: Journal of applied probability. Volume 58:Number 4(2021)
• Page Start: 1114
• Page End: 1130
• Publication Date: 2021-12
• Subjects: Condition numbers -- large deviations -- random matrices -- Wishart matrices; 60B20 -- 60F10 -- 15A12; 519.2
• DOI: 10.1017/jpr.2021.13
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: Let $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k ( p, n ) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k ( p, n ) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k ( p, n ) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.