Convergence of the empirical spectral distribution function of Beta matrices

TL;DR

This paper investigates the spectral distribution and fluctuations of Beta matrices, broadening applicability to cases where sample covariance matrices are not invertible, thus covering more practical scenarios in high-dimensional statistics.

Contribution

It establishes the limiting spectral distribution and central limit theorem for linear spectral statistics of Beta matrices without requiring invertibility of the component matrices.

Findings

Derived the limiting spectral distribution function.

Proved the central limit theorem for linear spectral statistics.

Applicable to high-dimensional cases where matrices are not invertible.

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.