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

2011/01/05 by Prathapasinghe Dharmawansa, Matthew R. McKay, Dharmawansa, Prathapasinghe +1
Computer Science · Mathematics · #33C15 #60B20 #62H10 #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Morphological variations and asymmetry #Random Matrices and Applications #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1101.1001

openalex publication_date 2011/01/05 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

Let \W be a correlated complex non-central Wishart matrix defined\nthrough \W=\X^H\X, where \X is n\× m\n , (n\≥ m) complex Gaussian with non-zero mean boldsymbol\Υ and\nnon-trivial covariance boldsymbol\Σ. We derive exact expressions for\nthe cumulative distribution functions (c.d.f.s) of the extreme eigenvalues\n(i.e., maximum and minimum) of \W for some particular cases. These\nresults are quite simple, involving rapidly converging infinite series, and\napply for the practically important case where boldsymbol\Υ has rank\none. We also derive analogous results for a certain class of gamma-Wishart\nrandom matrices, for which boldsymbol\Υ^H boldsymbol\Υ\nfollows a matrix-variate gamma distribution. The eigenvalue distributions in\nthis paper have various applications to wireless communication systems, and\narise in other fields such as econometrics, statistical physics, and\nmultivariate statistics.\n

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