Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix

TL;DR

This paper studies the spectral properties of the Moore-Penrose inverse of large sample covariance matrices, deriving limiting distributions and CLTs in high-dimensional settings where the number of variables exceeds the sample size.

Contribution

It provides the first limiting spectral distribution and CLT for the Moore-Penrose inverse of both centered and non-centered sample covariance matrices in high dimensions.

Findings

Marchenko-Pastur law applies to both matrices

Spectral distributions are the same for both matrices

Asymptotic means differ for linear spectral statistics

Abstract

For a sample of $n$ independent identically distributed $p$-dimensional centered random vectors with covariance matrix $\mathbf{\Sigma}_n$ let $\tilde{\mathbf{S}}_n$ denote the usual sample covariance (centered by the mean) and $\mathbf{S}_n$ the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where $p> n$. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of $\mathbf{S}_n$ and $\tilde{\mathbf{S}}_n$. We consider the large dimensional asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ such that $p/n\rightarrow c\in (1, +\infty)$. We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of $\tilde{\mathbf{S}}_n$ differs in the mean from that of $\mathbf{S}_n$.