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February, 1978 The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements
Kenneth W. Wachter
Ann. Probab. 6(1): 1-18 (February, 1978). DOI: 10.1214/aop/1176995607


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.


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Kenneth W. Wachter. "The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements." Ann. Probab. 6 (1) 1 - 18, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0374.60039
MathSciNet: MR467894
Digital Object Identifier: 10.1214/aop/1176995607

Primary: 60F15
Secondary: 62H25

Keywords: limit distributions , principal components , Random matrix spectra , singular values

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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