No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional noncentral Sample Covariance Matrices

2023/03/22 by Bai, Zhidong, Hu, Jiang, Silverstein, Jack W. +1 · 1 citation
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2303.12478

Abstract

Let \bbB_n =(1)/(n)(\bbR_n + \bbT^1/2_n \bbX_n)(\bbR_n + \bbT^1/2_n \bbX_n)^* , where \bbX_n is a p × n matrix with independent standardized random variables, \bbR_n is a p × n non-random matrix and \bbT_n is a p × p non-random, nonnegative definite Hermitian matrix. The matrix \bbB_n is referred to as the information-plus-noise type matrix, where \bbR_n contains the information and \bbT^1/2_n \bbX_n is the noise matrix with the covariance matrix \bbT_n . It is known that, as n → ∞ , if p/n converges to a positive number, the empirical spectral distribution of \bbB_n converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of \bbR_n and \bbT_n , for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all n sufficiently large.

Cited by

• No Eigenvalues Outside the Limiting Support of Generally Correlated and Noncentral Sample Covariance Matrices

Related

• Exact Separation of Eigenvalues of Large Dimensional Noncentral Sample Covariance Matrices
• No eigenvalues outside the limiting support of the spectral distribution of general sample covariance matrices
• The limiting spectral distribution of large dimensional general information-plus-noise type matrices
• Analysis of the limiting spectral distribution of large dimensional General information-plus-noise type matrices
• Extremal eigenvalues of sample covariance matrices with general\n population