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 $${\mathcal{L}}_n=\sum _{\alpha =1}^m{\xi }_{\alpha }{\mathbf{y}}_{\alpha }{\mathbf{y}}_{\alpha }^T\text{and}{\mathcal{A}}_n=\sum _{\alpha =1}^m{\xi }_{\alpha }({\mathbf{y}}_{\alpha }{\mathbf{x}}_{\alpha }^T+{\mathbf{x}}_{\alpha }{\mathbf{y}}_{\alpha }^T),$$ where for every $n$, $(\xi_\alpha )\alpha \subset $\mathbb{R}$$areiidrandomvariablesindependentof$($\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 $n,m(n)\to \mathrm{\infty }$: a standard one, where $m\mathrm{/}n\to c$, and a slightly modified one, where $m\mathrm{/}n\to \mathrm{\infty }$ and $\mathbf{E}\xi \to 0$ while $m\mathbf{E}\xi \mathrm{/}n\to c$ 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].