Spectral approximation for the separable covariance mixture model

Abstract

This paper introduces the separable covariance mixture model, which assumes a data-matrix $Y$ to be of the form $$\sum _{r=1}^R{A}_{r}X{B}_r$$ 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 $(\frac{1}{n}Y{Y}^{\ast }-z{\mathrm{Id}}_d{)}^{-1}$ and $(\frac{1}{n}{Y}^{\ast }Y-z{\mathrm{Id}}_n{)}^{-1}$ respectively approximate the deterministic matrices $$-\frac{1}{z}({\mathrm{Id}}_d+\sum _{r,s=1}^R\hat{I}{´}_{r,s}^{(B)}(z)A_r{A}_s^{\ast }{)}^{-1}\text{ and }-\frac{1}{z}({\mathrm{Id}}_n+\sum _{r,s=1}^R\hat{I}{´}_{r,s}^{(A)}(z)B_s^{\ast }{B}_r{)}^{-1},$$ where $\hat{I}{´}^{(A)},\hat{I}{´}^{(B)}\in {\mathbb{C}}^{R\times R}$ 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 $(A_r{)}_{r\le R}$ or $(B_r{)}_{r\le R}$, 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 $\frac{1}{n}Y{Y}^{\ast }$ or $\frac{1}{n}{Y}^{\ast }Y$, is included.

96 pages, 2 figures