The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements

Abstract

This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The limit is the limit as both dimensions grow large in some ratio. The matrix elements are required to have uniformly bounded central $2 + \delta$th moments, and the same means and variances within a row. The first section (relaxing the restriction on variances) proves any limit-in-distribution to be a constant measure rather than a random measure, establishes the existence of subsequences convergent in probability, and gives a criterion for almost-sure convergence. The second section proves the almost-sure limit to exist whenever the distribution of the row variances converges. It identifies the limit as a nonrandom probability measure which may be evaluated as a function of the limiting distribution of row variances and the dimension ratio. These asymptotic formulae underlie recently developed methods of probability plotting for principal components and have applications to multiple discriminant ratios and other linear multivariate statistics.