Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices

Abstract

Let $X$ be a $p\times n$ independent identically distributed real Gaussian matrix with positive mean $\mu $ and variance $\sigma^2$ entries. The goal of this paper is to investigate the largest eigenvalue of the noncentral sample covariance matrix $W=XX^{T}/n$, when the dimension $p$ and the sample size $n$ both grow to infinity with the limit $p/n=c\,(0<c<\infty)$. Utilizing the von Mises iteration method, we derive an approximation of the largest eigenvalue $\lambda_{1}(W)$ and show that $\lambda_{1}(W)$ asymptotically has a normal distribution with expectation $p\mu^2+(1+c)\sigma^2$ and variance $4c\mu^2\sigma^2$.