Singular values of large non-central random matrices

TL;DR

This paper analyzes the asymptotic behavior of the largest singular values of large non-central random matrices with fixed-rank means, providing a limit theorem and normality results for applications like population genetics.

Contribution

It introduces a limit theorem for the largest singular values of large non-central matrices with fixed-rank means, including explicit fluctuation representations and normality results.

Findings

Decomposition of largest singular values into deterministic and stochastic parts

Asymptotic normality of the largest singular values in block-structured matrices

Application to eigenvalues in population genetics models

Abstract

We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest $K$ singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of the normalized covariance matrix arising in a model of population genetics.