Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with zero mean and unit variance, and let 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λn < ∞ denote the eigenvalues of A * A, where (·) * represents conjugate-transpose. This paper investigates the distribution of the random variables n j=1 λ j λ k for k = 1 and k = 2. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as n and m tend to infinity with their difference fixed, both densities scale on the order of n 3 . After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case m = n.
Abstract-An exact expression for the joint density of three correlated Rician variables is not available in the open literature. In this letter, we derive new infinite series representations for the trivariate Rician probability density function (pdf) and the joint cumulative distribution function (cdf). Our results are limited to the case where the inverse covariance matrix is tridiagonal. This case seems the most general one that is tractable with Miller's approach and cannot be extended to more than three Rician variables. The outage probability of triple branch selective combining (SC) receiver over correlated Rician channels is presented as an application of the density function.
In this paper, we derive a new infinite series representation for the trivariate Non-central chi-squared distribution when the underlying correlated Gaussian variables have tridiagonal form of inverse covariance matrix. We make use of the Miller's approach and the Dougall's identity to derive the joint density function. Moreover, the trivariate cumulative distribution function (cdf) and characteristic function (chf) are also derived. Finally, bivariate noncentral chi-squared distribution and some known forms are shown to be special cases of the more general distribution. However, noncentral chi-squared distribution for an arbitrary covariance matrix seems intractable with the Miller's approach.
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