Unbounded Largest Eigenvalue of Large Sample Covariance Matrices: Asymptotics, Fluctuations and Applications

Abstract

Given a large sample covariance matrix $S_N=\frac 1n\Gamma_N^{1/2}Z_N Z_N^*\Gamma_N^{1/2}\, ,$ where $Z_N$ is a $N\times n$ matrix with i.i.d. centered entries, and $\Gamma_N$ is a $N\times N$ deterministic Hermitian positive semidefinite matrix, we study the location and fluctuations of $\lambda_{\max}(S_N)$, the largest eigenvalue of $S_N$ as $N,n\to\infty$ and $Nn^{-1} \to r\in(0,\infty)$ in the case where the empirical distribution $\mu^{\Gamma_N}$ of eigenvalues of $\Gamma_N$ is tight (in $N$) and $\lambda_{\max}(\Gamma_N)$ goes to $+\infty$. These conditions are in particular met when $\mu^{\Gamma_N}$ weakly converges to a probability measure with unbounded support on $\mathbb{R}^+$. We prove that asymptotically $\lambda_{\max}(S_N)\sim \lambda_{\max}(\Gamma_N)$. Moreover when the $\Gamma_N$'s are block-diagonal, and the following {\em spectral gap condition} is assumed:$$\limsup_{N\to\infty} \frac{\lambda_2(\Gamma_N)}{\lambda_{\max}(\Gamma_N)}<1,$$where $\lambda_2(\Gamma_N)$ is the second largest eigenvalue of $\Gamma_N$, we prove Gaussian fluctuations for $\lambda_{\max}(S_N)/\lambda_{\max}(\Gamma_N)$ at the scale $\sqrt{n}$.In the particular case where $Z_N$ has i.i.d. Gaussian entries and $\Gamma_N$ is the $N\times N$ autocovariance matrix of a long memory Gaussian stationary process $({\mathcal X}_t)_{t\in\mathbb{Z}}$, the columns of $\Gamma_N^{1/2} Z_N$ can be considered as $n$ i.i.d. samples of the random vector $({\mathcal X}_1,\dots,{\mathcal X}_N)^T$. We then prove that $\Gamma_N$ is similar to a diagonal matrix which satisfies all the required assumptions of our theorems, hence our results apply to this case.