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Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

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  • Dharmawansa, Prathapasinghe

Abstract

Let W be an n×n complex non-central Wishart matrix with m(≥n) degrees of freedom and a rank-1 mean. In this paper, we consider three problems related to the eigenvalues of W. To be specific, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue (λmin) of W. The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference m−n. This further facilitates the analysis of the microscopic limit of the minimum eigenvalue. The microscopic limit takes the form of the determinant of a square matrix with its entries expressed in terms of the modified Bessel functions of the first kind. We also develop a moment generating function based approach to derive the probability density function of the random variable tr(W)/λmin, where tr(⋅) denotes the trace of a square matrix. Moreover, we establish that, as m,n→∞ with m−n fixed, tr(W)/λmin scales like n3. Finally, we find the average of the reciprocal of the characteristic polynomial det[zIn+W],|argz|<π, where In and det[⋅] denote the identity matrix of size n and the determinant, respectively.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:149:y:2016:i:c:p:30-53
    DOI: 10.1016/j.jmva.2016.03.003
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    References listed on IDEAS

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    1. Krishnaiah, P. R. & Schuurmann, F. J., 1974. "On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 4(3), pages 265-282, September.
    2. 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.
    3. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, September.
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