Extreme Eigenvalue Distributions of Some Complex Correlated Non-Central Wishart and Gamma-Wishart Random Matrices

Abstract

Let $\mathbf{W}$ be a correlated complex non-central Wishart matrix defined through $\mathbf{W}=\mathbf{X}^H\mathbf{X}$, where $\mathbf{X}$ is $n\times m \, (n\geq m)$ complex Gaussian with non-zero mean $\boldsymbol{\Upsilon}$ and non-trivial covariance $\boldsymbol{\Sigma}$. We derive exact expressions for the cumulative distribution functions (c.d.f.s) of the extreme eigenvalues (i.e., maximum and minimum) of $\mathbf{W}$ for some particular cases. These results are quite simple, involving rapidly converging infinite series, and apply for the practically important case where $\boldsymbol{\Upsilon}$ has rank one. We also derive analogous results for a certain class of gamma-Wishart random matrices, for which $\boldsymbol{\Upsilon}^H\boldsymbol{\Upsilon}$ follows a matrix-variate gamma distribution. The eigenvalue distributions in this paper have various applications to wireless communication systems, and arise in other fields such as econometrics, statistical physics, and multivariate statistics.