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

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