{"ID":2890961,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.18505","arxiv_id":"2507.18505","title":"LSD of sample covariances of superposition of matrices with separable covariance structure","abstract":"We study the asymptotic behavior of the spectra of matrices of the form $S_n = \\frac{1}{n}XX^*$ where $X =\\sum_{r=1}^K X_r$, where $X_r = A_r^\\frac{1}{2}Z_rB_r^\\frac{1}{2}$, $K \\in \\mathbb{N}$ and $A_r,B_r$ are sequences of positive semi-definite matrices of dimensions $p\\times p$ and $n\\times n$, respectively. We establish the existence of a limiting spectral distribution for $S_n$ by assuming that matrices $\\{A_r\\}_{r=1}^K$ are simultaneously diagonalizable and $\\{B_r\\}_{r=1}^K$ are simultaneously digaonalizable, and that the joint spectral distributions of $\\{A_r\\}_{r=1}^K$ and $\\{B_r\\}_{r=1}^K$ converge to $K$-dimensional distributions, as $p,n\\to \\infty$ such that $p/n \\to c \\in (0,\\infty)$. The LSD of $S_n$ is characterized by system of equations with unique solutions within the class of Stieltjes transforms of measures on $\\mathbb{R}_+$. These results generalize existing results on the LSD of sample covariances when the data matrices have a separable covariance structure.","short_abstract":"We study the asymptotic behavior of the spectra of matrices of the form $S_n = \\frac{1}{n}XX^*$ where $X =\\sum_{r=1}^K X_r$, where $X_r = A_r^\\frac{1}{2}Z_rB_r^\\frac{1}{2}$, $K \\in \\mathbb{N}$ and $A_r,B_r$ are sequences of positive semi-definite matrices of dimensions $p\\times p$ and $n\\times n$, respectively. We esta...","url_abs":"https://arxiv.org/abs/2507.18505","url_pdf":"https://arxiv.org/pdf/2507.18505v2","authors":"[\"Javed Hazarika\",\"Debashis Paul\"]","published":"2025-07-24T15:22:38Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}