Convergence of the empirical spectral distribution function of Beta matrices

Abstract

Let ${\mathbf{B}}_n={\mathbf{S}}_n({\mathbf{S}}_n+{\alpha }_n{\mathbf{T}}_N{)}^{-1}$, where ${\mathbf{S}}_n$ and ${\mathbf{T}}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of ${\mathbf{B}}_n$. Especially, we do not require ${\mathbf{S}}_n$ or ${\mathbf{T}}_N$ to be invertible. Namely, we can deal with the case where $p>\max \{n,N\}$ and $p<n+N$. Therefore, our results cover many important applications which cannot be simply deduced from the corresponding results for multivariate $F$ matrices.

Published at http://dx.doi.org/10.3150/14-BEJ613 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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