LSD of sample covariances of superposition of matrices with separable covariance structure

2025/07/24 by Hazarika, Javed, Paul, Debashis
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2507.18505

Abstract

We study the asymptotic behavior of the spectra of matrices of the form S_n = (1)/(n)XX^* where X =∑_r=1^K X_r, where X_r = A_r^(1)/(2)Z_rB_r^(1)/(2), K ∈ ℕ and A_r,B_r are sequences of positive semi-definite matrices of dimensions p× p and n× 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→ ∞ such that p/n → c ∈ (0,∞). The LSD of S_n is characterized by system of equations with unique solutions within the class of Stieltjes transforms of measures on ℝ₊. These results generalize existing results on the LSD of sample covariances when the data matrices have a separable covariance structure.

Citations

• Separable Expansions for Covariance Estimation
• A Test for Separability in Covariance Operators of Random Surfaces
• Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices
• Testing separability of space--time functional processes
• Cross-Covariance Functions for Multivariate Geostatistics
• Analysis of the limiting spectral measure of large random matrices of the separable covariance type
• Limiting spectral distribution of renormalized separable sample covariance matrices when p/n→ 0
• Limiting spectral distribution of renormalized separable sample covariance matrices whenp/n→0
• On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

Related

• Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices
• XX^T Matrices With Independent Entries
• On singular values of data matrices with general independent columns
• Limiting spectral distribution of renormalized separable sample covariance matrices when p/n→ 0
• Linear spectral statistics of sequential sample covariance matrices