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| Formato: | Preprint |
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2026
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| Acceso en línea: | https://arxiv.org/abs/2604.18181 |
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| _version_ | 1866913050622164992 |
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| author | Deitmar, Ben |
| author_facet | Deitmar, Ben |
| contents | This paper introduces the separable covariance mixture model, which assumes a data-matrix $Y$ to be of the form $$ \sum\limits_{r=1}^R A_r X B_r $$ for one random $(d \times n)$-matrix $X$ with independent centered variance-one entries, and for two families of deterministic matrices $A_1,\dots,A_R \in \mathbb{C}^{d \times d}$ and $B_1,\dots,B_R \in \mathbb{C}^{n \times n}$. Under certain assumptions, it is shown that the resolvents $(\frac{1}{n} Y Y^* - z \operatorname{Id}_d)^{-1}$ and $(\frac{1}{n} Y^* Y - z \operatorname{Id}_n)^{-1}$ respectively approximate the deterministic matrices $$
-\frac{1}{z}\Big( \operatorname{Id}_d + \sum\limits_{r,s=1}^R δ^{(B)}_{r,s}(z) A_{r} A_{s}^* \Big)^{-1} \ \ \text{ and } \ \ -\frac{1}{z}\Big( \operatorname{Id}_n + \sum\limits_{r,s=1}^R δ^{(A)}_{r,s}(z) B_{s}^*B_{r} \Big)^{-1} \ , $$ where $δ^{(A)}, δ^{(B)} \in \mathbb{C}^{R \times R}$ are uniquely defined solutions to a certain dual system of equations. The results are non-asymptotic and do not require simultaneous diagonalizability of the families $(A_r)_{r \leq R}$ or $(B_r)_{r \leq R}$, as was required in previous works such as [Hazarika and Paul (2025)] or [Mei et al. (2023)]. An asymptotic application, which describes the limiting spectral distribution of the sample covariance matrix analogues $\frac{1}{n} Y Y^*$ or $\frac{1}{n} Y^* Y$, is included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral approximation for the separable covariance mixture model Deitmar, Ben Statistics Theory This paper introduces the separable covariance mixture model, which assumes a data-matrix $Y$ to be of the form $$ \sum\limits_{r=1}^R A_r X B_r $$ for one random $(d \times n)$-matrix $X$ with independent centered variance-one entries, and for two families of deterministic matrices $A_1,\dots,A_R \in \mathbb{C}^{d \times d}$ and $B_1,\dots,B_R \in \mathbb{C}^{n \times n}$. Under certain assumptions, it is shown that the resolvents $(\frac{1}{n} Y Y^* - z \operatorname{Id}_d)^{-1}$ and $(\frac{1}{n} Y^* Y - z \operatorname{Id}_n)^{-1}$ respectively approximate the deterministic matrices $$ -\frac{1}{z}\Big( \operatorname{Id}_d + \sum\limits_{r,s=1}^R δ^{(B)}_{r,s}(z) A_{r} A_{s}^* \Big)^{-1} \ \ \text{ and } \ \ -\frac{1}{z}\Big( \operatorname{Id}_n + \sum\limits_{r,s=1}^R δ^{(A)}_{r,s}(z) B_{s}^*B_{r} \Big)^{-1} \ , $$ where $δ^{(A)}, δ^{(B)} \in \mathbb{C}^{R \times R}$ are uniquely defined solutions to a certain dual system of equations. The results are non-asymptotic and do not require simultaneous diagonalizability of the families $(A_r)_{r \leq R}$ or $(B_r)_{r \leq R}$, as was required in previous works such as [Hazarika and Paul (2025)] or [Mei et al. (2023)]. An asymptotic application, which describes the limiting spectral distribution of the sample covariance matrix analogues $\frac{1}{n} Y Y^*$ or $\frac{1}{n} Y^* Y$, is included. |
| title | Spectral approximation for the separable covariance mixture model |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2604.18181 |