Eigenvalue statistics for the sum of two complex Wishart matrices

TL;DR

This paper derives exact eigenvalue statistics for the sum of two complex Wishart matrices with specific covariance structures, filling a gap in multivariate statistical analysis and related applications.

Contribution

It provides the first analytical solutions for eigenvalue distributions of the sum of two Wishart matrices with unequal covariances, using a generalized integral approach.

Findings

Exact joint and marginal eigenvalue densities derived

Analytical results match numerical simulations perfectly

Applicable to multivariate statistics, finance, and telecommunications

Abstract

The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement.