%0 Journal Article
%T Convergence of the empirical spectral distribution function of Beta matrices
%A Zhidong Bai
%A Jiang Hu
%A Guangming Pan
%A Wang Zhou
%J Mathematics
%D 2012
%I arXiv
%R 10.3150/14-BEJ613
%X 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.
%U http://arxiv.org/abs/1208.5953v3