Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

  • Authors:

  • Guangming Pan

  • Affiliations:

  • Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2010

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Abstract

Let B"n=A"n+X"nT"nX"n^T, where A"n is a random symmetric matrix, T"n a random symmetric matrix, and X"n=1n(X"i"j^(^n^))"n"x"p with X"i"j^(^n^) being independent real random variables. Suppose that X"n, T"n and A"n are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices B"n converges almost surely to a non-random distribution.