Asymptotic Normality of the Largest Eigenvalue for Noncentral Sample Covariance Matrices

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 $W=X{X}^T\mathrm{/}n$, when the dimension $p$ and the sample size $n$ both grow to infinity with the limit $p\mathrm{/}n=c(0<c<\mathrm{\infty })$. 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 $4c{\mu }^2{\sigma }^2$.