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This paper derives central limit theorems (CLTs) for general linear spectralstatistics (LSS) of three important multi-spiked Hermitian random matrixensembles. The first is the most common spiked scenario, proposed by Johnstone,which is a central Wishart ensemble with fixed-rank perturbation of theidentity matrix, the second is a non-central Wishart ensemble with fixed-ranknoncentrality parameter, and the third is a similarly defined non-central $F$ensemble. These CLT results generalize our recent work to account for multiplespikes, which is the most common scenario met in practice. The generalizationis non-trivial, as it now requires dealing with hypergeometric functions ofmatrix arguments. To facilitate our analysis, for a broad class of suchfunctions, we first generalize a recent result of Onatski to present newcontour integral representations, which are particularly suitable for computinglarge-dimensional properties of spiked matrix ensembles. Armed with suchrepresentations, our CLT formulas are derived for each of the three spikedmodels of interest by employing the Coulomb fluid method from random matrixtheory along with saddlepoint techniques. We find that for each matrix model,and for general LSS, the individual spikes contribute additively to yield a$O(1)$ correction term to the asymptotic mean of the linear statistic, which wespecify explicitly, whilst having no effect on the leading order terms of themean or variance.

Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models