Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices

TL;DR

This paper establishes that the largest eigenvalue of noncentral sample covariance matrices from Gaussian data converges to a normal distribution as dimensions grow, providing explicit formulas for its mean and variance.

Contribution

It derives the asymptotic normality of the largest eigenvalue for noncentral covariance matrices using von Mises iteration, a novel approach in this context.

Findings

Largest eigenvalue is asymptotically normal

Explicit formulas for mean and variance

Applicable for high-dimensional Gaussian data

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 $4c\mu^2\sigma^2$.