Large-deviation asymptotics of condition numbers of random matrices

• Published in: Journal of applied probability 2021-12, Vol.58 (4), p.1114-1130
• Main Authors: Singull, Martin, Uwamariya, Denise, Yang, Xiangfeng
• Format: Article
• Language: English
• Subjects: Condition numbers; Deviation; Eigenvalues; Euclidean space; Hypotheses; Independent variables; Investigations; large deviations; Linear algebra; Normal distribution; Original Article; Population; random matrices; Random variables; Real variables; Statistical analysis; Statistical inference; Test methods; Wishart matrices
• ISSN: 0021-9002; 1475-6072; 1475-6072
• DOI: 10.1017/jpr.2021.13

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.