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.