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Large-deviation asymptotics of condition numbers of random matrices

Part of: Limit theorems Basic linear algebra Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  22 November 2021

Martin Singull ,
Denise Uwamariya  and
Xiangfeng Yang
Martin Singull*
Affiliation:
Linköping University
Denise Uwamariya*
Affiliation:
Linköping University
Xiangfeng Yang*
Affiliation:
Linköping University
*
*Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
*Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
*Postal address: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden.
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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.

Keywords

Condition numberslarge deviationsrandom matricesWishart matrices

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Secondary: 60F10: Large deviations 15A12: Conditioning of matrices
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1114 - 1130
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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