Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices

Abstract

Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three “spiked” Hermitian random matrix ensembles. These include Johnstone’s spiked model (i.e., central Wishart with spiked correlation), non-central Wishart with rank-one non-centrality, and a related class of non-central $$F$$ matrices. For a generic linear statistic, we derive simple and explicit CLT expressions as the matrix dimensions grow large. For all three ensembles under consideration, we find that the primary effect of the spike is to introduce an $$O(1)$$ correction term to the asymptotic mean of the linear spectral statistic, which we characterize with simple formulas. The utility of our proposed framework is demonstrated through application to three different linear statistics problems: the classical likelihood ratio test for a population covariance, the capacity analysis of multi-antenna wireless communication systems with a line-of-sight transmission path, and a classical multiple sample significance testing problem.