Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices

Abstract

Let W be a correlated complex non-central Wishart matrix defined through W = X H X , where X is an n × m ( n ≥ m ) complex Gaussian with non-zero mean Υ and non-trivial covariance Σ . We derive exact expressions for the cumulative distribution functions (c.d.f.s) of the extreme eigenvalues (i.e., maximum and minimum) of W for some particular cases. These results are quite simple, involving rapidly converging infinite series, and apply for the practically important case where Υ has rank one. We also derive analogous results for a certain class of gamma-Wishart random matrices, for which Υ H Υ 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.