No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix

We consider a class of matrices of the form C n = ( 1 / N ) A n 1 / 2 X n B n X n ∗ × A n 1 / 2 , where X n is an n × N matrix consisting of i.i.d. standardized complex entries, A n 1 / 2 is a nonnegative definite square root of the nonnegative definite Hermitian matrix A n , and B n is diagonal wit...

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Bibliographic Details
Published in:Journal of multivariate analysis 2009, Vol.100 (1), p.37-57
Main Authors: Paul, Debashis, Silverstein, Jack W.
Format: Article
Language:English
Subjects:
60F20
60F20 62H99 Empirical spectral distribution Stieltjes transform Separable covariance CDMA MIMO
62H99
CDMA
Convergence
Distribution theory
Eigenvalues
Empirical spectral distribution
Exact sciences and technology
Limit theorems
Mathematics
Matrix
MIMO
Multivariate analysis
Nonparametric inference
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Separable covariance
Statistics
Stieltjes transform
Studies
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Summary:We consider a class of matrices of the form C n = ( 1 / N ) A n 1 / 2 X n B n X n ∗ × A n 1 / 2 , where X n is an n × N matrix consisting of i.i.d. standardized complex entries, A n 1 / 2 is a nonnegative definite square root of the nonnegative definite Hermitian matrix A n , and B n is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of A n and B n converge to proper probability distributions as n N → c ∈ ( 0 , ∞ ) , the empirical spectral distribution of C n converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of A n and B n , with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n . The problem is motivated by applications in spatio-temporal statistics and wireless communications.
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2008.03.010