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

Javed Hazarika, Debashis Paul

TL;DR

This work analyzes the limiting spectral distribution (LSD) of the sample covariance S_n = \frac{1}{n}XX^*, where X = \sum_{r=1}^K X_r and X_r = A_r^{1/2} Z_r B_r^{1/2}. Under simultaneous diagonalizability of the scaling matrices and convergence of their joint spectral distributions to H and G with p/n \to c, the LSD exists and is characterized by a system of integral equations for the Stieltjes transforms, with a unique solution in the Stieltjes class. The authors provide multiple equivalent characterizations of the LSD (via h and g fixed-point equations and via s_F(z)) and extend the results to non-diagonal scalings through a Lindeberg-type argument, as well as to general (potentially unbounded) population spectra. Special cases recover known separable-covariance results and yield closed-form expressions akin to the MP law under certain symmetry assumptions. The theoretical findings are complemented by simulations illustrating the agreement between empirical and predicted spectra and highlighting the impact of tail behavior on edge stability.

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.

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

TL;DR

This work analyzes the limiting spectral distribution (LSD) of the sample covariance S_n = \frac{1}{n}XX^*, where X = \sum_{r=1}^K X_r and X_r = A_r^{1/2} Z_r B_r^{1/2}. Under simultaneous diagonalizability of the scaling matrices and convergence of their joint spectral distributions to H and G with p/n \to c, the LSD exists and is characterized by a system of integral equations for the Stieltjes transforms, with a unique solution in the Stieltjes class. The authors provide multiple equivalent characterizations of the LSD (via h and g fixed-point equations and via s_F(z)) and extend the results to non-diagonal scalings through a Lindeberg-type argument, as well as to general (potentially unbounded) population spectra. Special cases recover known separable-covariance results and yield closed-form expressions akin to the MP law under certain symmetry assumptions. The theoretical findings are complemented by simulations illustrating the agreement between empirical and predicted spectra and highlighting the impact of tail behavior on edge stability.

Abstract

We study the asymptotic behavior of the spectra of matrices of the form where , where , and are sequences of positive semi-definite matrices of dimensions and , respectively. We establish the existence of a limiting spectral distribution for by assuming that matrices are simultaneously diagonalizable and are simultaneously digaonalizable, and that the joint spectral distributions of and converge to -dimensional distributions, as such that . The LSD of is characterized by system of equations with unique solutions within the class of Stieltjes transforms of measures on . These results generalize existing results on the LSD of sample covariances when the data matrices have a separable covariance structure.

Paper Structure

This paper contains 26 sections, 45 theorems, 199 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some established results
  3. Model and preliminaries
  4. L.S.D. under commutativity of scaling matrices
  5. Sketch of the proof
  6. Assumptions
  7. Extending to non-diagonal scaling matrices
  8. Existence of Solution under general conditions
  9. A few properties of the L.S.D.
  10. A few Special Cases
  11. Relaxation of commutativity requirement
  12. Simulation Study
  13. Some basic results
  14. Existence of solution under A1-A2
  15. Proof of Theorem \ref{['DeterministicEquivalent']}
  16. ...and 11 more sections

Key Result

Theorem 2.1

Suppose that ($P_n$) are real Borel probability measures with Stieltjes transforms ($S_n$) respectively. If $\underset{n\rightarrow\infty}{\lim}S_n(z)=S(z)$ for all z with $\Im(z) > 0$, then there exists a Borel probability measure $P$ with Stieltjes transform $S_P=S$ if and only if in which case $P_n\rightarrow P$ in distribution.

Figures (3)

  • Figure 1: Point Mass of the L.S.D. at $0$ as a function of $c$
  • Figure 2: Example 1: Simulated vs. Theoretical limit distributions for various values of $K$ and $c$ with standard Gaussian innovations.
  • Figure 3: Example 2: Simulated vs. Theoretical limit distributions for various values of $K$ and $c$ when the innovations follow $t$-distribution with $3$ degrees of freedom.

Theorems & Definitions (103)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 93 more