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

Abstract

Given a large sample covariance matrix $S_N=\frac{1}{n}{\Gamma }_N^{1/2}{Z}_N{Z}_N^{\ast }{\Gamma }_N^{1/2},$ 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 $N{n}^{-1}\to r\in (0,\infty )$ 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:$$\underset{N\to \infty }{\mathrm{limsup}}\frac{{\lambda }_2({\Gamma }_N)}{{\lambda }_{\max }({\Gamma }_N)}<1,$$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.

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