Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model

TL;DR

This paper extends the Wishart model to algebraic distributions for doubly correlated real covariance matrices, explicitly calculating matrix moments and addressing the mathematical challenges involved.

Contribution

It introduces a new algebraic generalization of the Wishart model for doubly correlated matrices and derives explicit formulas for the first two matrix moments.

Findings

Explicit calculation of the first and second matrix moments.

Relation of the problem to the Aomoto integral and Ingham-Siegel integrals.

Comparison with the Gaussian Wishart model.

Abstract

The Wishart model of random covariance or correlation matrices continues to find ever more applications as the wealth of data on complex systems of all types grows. The heavy tails often encountered prompt generalizations of the Wishart model, involving algebraic distributions instead of a Gaussian. The mathematical properties pose new challenges, particularly for the doubly correlated versions. Here we investigate such a doubly correlated algebraic model for real covariance or correlation matrices. We focus on the matrix moments and explicitly calculate the first and the second one, the computation of the latter is non-trivial. We solve the problem by relating it to the Aomoto integral and by extending the recursive technique to calculate Ingham-Siegel integrals. We compare our results with the Gaussian case.