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
