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

Given a large sample covariance matrix $S_N=\frac 1nΓ_N^{1/2}Z_N Z_N^*Γ_N^{1/2}\, ,$ where $Z_N$ is a $N\times n$ matrix with i.i.d. centered entries, and $Γ_N$ is a $N\times N$ deterministic Hermitian positive semidefinite matrix, we study the location and fluctuations of $λ_{\max}(S_N)$, 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 $μ^{Γ_N}$ of eigenvalues of $Γ_N$ is tight (in $N$) and $λ_{\max}(Γ_N)$ goes to $+\infty$. These conditions are in particular met when $μ^{Γ_N}$ weakly converges to a probability measure with unbounded support on $\mathbb{R}^+$. We prove that asymptotically $λ_{\max}(S_N)\sim λ_{\max}(Γ_N)$. Moreover when the $Γ_N$'s are block-diagonal, and the following {\em spectral gap condition} is assumed:$\limsup_{N\to\infty} \frac{λ_2(Γ_N)}{λ_{\max}(Γ_N)}<1,$where $λ_2(Γ_N)$ is the second largest eigenvalue of $Γ_N$, we prove Gaussian fluctuations for $λ_{\max}(S_N)/λ_{\max}(Γ_N)$ at the scale $\sqrt{n}$.In the particular case where $Z_N$ has i.i.d. Gaussian entries and $Γ_N$ is the $N\times N$ autocovariance matrix of a long memory Gaussian stationary process $({\mathcal X}_t)_{t\in\mathbb{Z}}$, the columns of $Γ_N^{1/2} Z_N$ 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 $Γ_N$ 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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