Asymptotic Joint Distribution of Extreme Eigenvalues of the Sample Covariance Matrix in the Spiked Population Model

TL;DR

This paper analyzes the asymptotic joint distribution of the largest and smallest eigenvalues of sample covariance matrices in a spiked population model, revealing phase transitions and asymptotic independence.

Contribution

It establishes the asymptotic distribution of extreme eigenvalues and demonstrates their asymptotic independence in the spiked covariance model.

Findings

Asymptotic distribution of the smallest eigenvalue derived.

Phase transition between Tracy-Widom and Gaussian fluctuations identified.

Largest and smallest eigenvalues are asymptotically independent.

Abstract

In this paper, we consider a data matrix $X\in\mathbb{C}^{N\times M}$ where all the columns are i.i.d. samples being $N$ dimensional complex Gaussian of mean zero and covariance $\Sigma\in\mathbb{C}^{N\times N}$. Here the population matrix $\Sigma$ is of finite rank perturbation of the identity matrix. This is the "spiked population model" first proposed by Johnstone in \cite{21}. As $M, N\to\infty$ but $M/N = \gamma\in(1, \infty)$, we first establish in this paper the asymptotic distribution of the smallest eigenvalue of the sample covariance matrix $S:= XX^*/M$. It also exhibits a phase transition phenomenon proposed in \cite{1} --- the local fluctuation will be the generalized Tracy-Widom or the generalized Gaussian to be defined in the paper. Moreover we prove that the largest and the smallest eigenvalue are asymptotically independent as $M, N\to\infty$.