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

Given a large sample covariance matrix $S_N=\frac 1n\Gamma_N^{1/2}Z_N Z_N^*\Gamma_N^{1/2}, ,$where$Z_N$isa$N\times n$ matrix with i.i.d. centered entries, and ${\mathrm{\Gamma }}_N$ 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 $N,n\to \mathrm{\infty }$ and $N{n}^{-1}\to r\in (0,\mathrm{\infty })$ in the case where the empirical distribution ${\mu }^{{\mathrm{\Gamma }}_N}$ of eigenvalues of ${\mathrm{\Gamma }}_N$ is tight (in $N$) and ${\lambda }_{\max }({\mathrm{\Gamma }}_N)$ goes to $+\mathrm{\infty }$. These conditions are in particular met when ${\mu }^{{\mathrm{\Gamma }}_N}$ weakly converges to a probability measure with unbounded support on ${\mathbb{R}}^{+}$. We prove that asymptotically ${\lambda }_{\max }(S_N)\sim {\lambda }_{\max }({\mathrm{\Gamma }}_N)$. Moreover when the ${\mathrm{\Gamma }}_N$’s are block-diagonal, and the following {\em spectral gap condition} is assumed:$$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{{\lambda }_2({\mathrm{\Gamma }}_N)}{{\lambda }_{\max }({\mathrm{\Gamma }}_N)}<1,$$where ${\lambda }_2({\mathrm{\Gamma }}_N)$ is the second largest eigenvalue of ${\mathrm{\Gamma }}_N$, we prove Gaussian fluctuations for ${\lambda }_{\max }(S_N)\mathrm{/}{\lambda }_{\max }({\mathrm{\Gamma }}_N)$ at the scale $\sqrt{n}$.In the particular case where $Z_N$ has i.i.d. Gaussian entries and ${\mathrm{\Gamma }}_N$ is the $N\times N$ autocovariance matrix of a long memory Gaussian stationary process $({\mathcal{X}}_t{)}_{t\in \mathbb{Z}}$, the columns of ${\mathrm{\Gamma }}_N^{1\mathrm{/}2}{Z}_N$ 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 ${\mathrm{\Gamma }}_N$ is similar to a diagonal matrix which satisfies all the required assumptions of our theorems, hence our results apply to this case.