On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Abstract

Given $n,m\in \mathbb{N}$, we study two classes of large random matrices of the form $$ \mathcal{L}_n =\sum_{\alpha=1}^m\xi_\alpha \mathbf{y}_\alpha \mathbf{y}_\alpha ^T\quad\text{and}\quad \mathcal{A}_n =\sum_{\alpha =1}^m\xi_\alpha (\mathbf{y}_\alpha \mathbf{x}_\alpha ^T+\mathbf{x}_\alpha \mathbf{y}_\alpha ^T), $$ where for every $n$, $(\xi_\alpha )_\alpha \subset \mathbb{R}$ are iid random variables independent of $(\mathbf{x}_\alpha,\mathbf{y}_\alpha)_\alpha$, and $(\mathbf{x}_\alpha )_\alpha $, $(\mathbf{y}_\alpha )_\alpha \subset \mathbb{R}^n$ are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as $n,m(n)\to \infty$: a standard one, where $m/n\to c$, and a slightly modified one, where $m/n\to\infty$ and $\mathbf{E}\xi\to 0$ while $m\mathbf{E}\xi /n\to c$ for some $c\ge 0$. Assuming that vectors $(\mathbf{x}_\alpha )_\alpha $ and $(\mathbf{y}_\alpha )_\alpha $ are normalized and isotropic "in average", we prove the convergence in probability of the empirical spectral distributions of $\mathcal{L}_n $ and $\mathcal{A}_n $ to a version of the Marchenko-Pastur law and so called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as $(\xi_\alpha )_\alpha $, in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [9, 21].