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

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 $S:=X{X}^{\ast }/M$. 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 $M,N\to \infty$.

36 pages, 5 figures

References in corpus (1)

• Instabilities in complex mixtures with a large number of components