Alicja Dembczak-Kołodziejczyk

arXiv2024

A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices

In this note, we consider a sample covariance matrix of the form $$M_{n}=\sum_{α=1}^m τ_α{\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)}({\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)})^T,$$ where $(\mathbf{y}_α^{(1)},\, {\mathbf{y}}_α^{(2)})_α$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $τ_α\in \mathbb{R}_+ $. We show that as $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge η_0>0$, converge in distribution to a two-dimensional Gaussian random variable with zero mean and a certain covariance matrix. This work is a continuation of Dembczak-Kołodziejczyk and Lytova (2023), and Lytova (2018).

math.PR↗

arXiv2021

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].

math.PR↗