On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors

On the Distribution of Matrix Quadratic Forms

“…In (Khatri, 1977) the Laplace transform was used to generalize the results of Shah (1970) In this paper a characterization of the distribution of the quadratic form Q when X ∼ N p,n (M, Σ, Ψ) is given. Instead of representing it in terms of a hypergeometric function of matrix argument and an expansion in zonal polynomials as in (Khatri, 1966) and (Hayakawa, 1966) we show that the distribution of Q coincide with the distribution of a weighted sum of non-central Wishart distributed matrices, similar as in the case when M = 0 and Ψ = I done by Hayakawa (1966). We also discuss the complex normal case and show that the same properties hold.…”

“…Khatri (1962) Mathew and Nordström (1997); Masaro and Wong (2003) discussed Wishartness for the quadratic form when the covariance matrix is non-separable, i.e., when the covariance matrix cannot be written as a Kronecker product. Khatri (1966) derived the density for Q in the central case, i.e., when M = 0. The density function involves the hypergeometric function of matrix argument and is cumbersome to 2 handle.…”

“…The probability density function given by Khatri (1966) is written as the product of a Wishart density function and a generalized hypergeometric function. This form is not always convenient for studying properties of Q.…”

High-SNR power offset in multi-antenna communication

“…414 and 418], also into the closed form in (110). Using the following generalization to complex matrices of the result given by real matrices in [54] (126) the moment-generating function turns out to be (127) from which (128) (129) which, using (136), leads to the claim. 10 In fact, [54] gives the density distribution for matrices of the form from which, applying (132), the corresponding expression in Proposition 4 is found.…”

“…The excess power offset does become explicit whenever correlation is present only at the side of the link with the fewest antennas, in which case (54) As an alternative to the exact expressions, can be approximated by resorting to the large-dimensional results in Proposition 5 (with the expectations replaced by arithmetic means over the diagonal entries of and ) and Proposition 2. It is worth summarizing the insight provided by the power offset on the impact of correlation on the capacity at high , contrasting it with how it affects it at low [16], [32].…”