LSD of sample covariances of superposition of matrices with separable covariance structure
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arXiv
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2025
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| author | Hazarika, Javed Paul, Debashis |
| author_facet | Hazarika, Javed Paul, Debashis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | LSD of sample covariances of superposition of matrices with separable covariance structure Hazarika, Javed Paul, Debashis Statistics Theory 62H10, 60B20 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. |
| title | LSD of sample covariances of superposition of matrices with separable covariance structure |
| topic | Statistics Theory 62H10, 60B20 |
| url | https://arxiv.org/abs/2507.18505 |