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

TL;DR

This paper investigates the asymptotic behavior and fluctuations of the largest eigenvalue of large sample covariance matrices, especially when the population covariance matrix has unbounded spectral support, with applications to long memory processes.

Contribution

It establishes the asymptotic equivalence of the largest eigenvalue to the maximum eigenvalue of the population covariance, and proves Gaussian fluctuations under spectral gap conditions.

Findings

Largest eigenvalue asymptotically equals the maximum eigenvalue of the population matrix.

Gaussian fluctuations occur under spectral gap conditions.

Results apply to long memory Gaussian stationary processes.

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.