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

Given $n,m\in \mathbb{N}$, we study two classes of large random matrices of the form $\mathcal{L}_n =\sum_{α=1}^mξ_α\mathbf{y}_α\mathbf{y}_α^T\quad\text{and}\quad \mathcal{A}_n =\sum_{α=1}^mξ_α(\mathbf{y}_α\mathbf{x}_α^T+\mathbf{x}_α\mathbf{y}_α^T),$ where for every $n$, $(ξ_α)_α\subset \mathbb{R}$ are iid random variables independent of $(\mathbf{x}_α,\mathbf{y}_α)_α$, and $(\mathbf{x}_α)_α$, $(\mathbf{y}_α)_α\subset \mathbb{R}^n$ 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 $\mathbf{E}ξ\to 0$ while $m\mathbf{E}ξ/n\to c$ for some $c\ge 0$. Assuming that vectors $(\mathbf{x}_α)_α$ and $(\mathbf{y}_α)_α$ 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 $(ξ_α)_α$, 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].

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