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

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  • Dharmawansa, Prathapasinghe
  • McKay, Matthew R.

Abstract

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.

Suggested Citation

  • Dharmawansa, Prathapasinghe & McKay, Matthew R., 2011. "Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 847-868, April.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:4:p:847-868
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    References listed on IDEAS

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    1. C. Khatri, 1969. "Non-central distributions ofith largest characteristic roots of three matrices concerning complex multivariate normal populations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 23-32, December.
    2. Smith, Peter J. & Garth, Lee M., 2007. "Distribution and characteristic functions for correlated complex Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 661-677, April.
    3. Chikuse, Yasuko & Davis, A. W., 1986. "A Survey on the Invariant Polynomials with Matrix Arguments in Relation to Econometric Distribution Theory," Econometric Theory, Cambridge University Press, vol. 2(2), pages 232-248, August.
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    Cited by:

    1. Dharmawansa, Prathapasinghe, 2016. "Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 30-53.
    2. Daya K. Nagar & Alejandro Roldán-Correa & Saralees Nadarajah, 2023. "Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix," Mathematics, MDPI, vol. 11(9), pages 1-14, May.

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