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

  • Authors:

  • Debashis Paul;Jack W. Silverstein

  • Affiliations:

  • Department of Statistics, University of California, Davis, CA 95616, USA;Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205, USA and SAMSI, Research Triangle Park, NC 27709-4006, USA

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2009

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Abstract

We consider a class of matrices of the form C"n=(1/N)A"n^1^/^2X"nB"nX"n^*xA"n^1^/^2, where X"n is an nxN 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 nN-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.