Non-Central Multivariate Chi-Square and Gamma Distributions

Abstract

A 1 () p --variate integral representation is given for the cumulative distribution function of the general p -variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from ( , ).  To facilitate the computation, these formulas are given more detailed for 3. p  These 1 () p --variate integrals are also derived for the diagonal of a non-central complex Wishart matrix. Furthermore, some alternative formulas are given for the cases with an associated "one-factorial" () pp  -correlation matrix , R i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3 3)  -R with no vanishing correlation.

A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions

“…respectively with the rows b j from B and the expectation referring to S, a W n−1 (2α, I n−1 )-Wishart matrix. These formulas are derived in a slightly more general form in [10] and [11]; they can be verified by a simple calculation using the Laplace transform of the non-central gamma distribution followed by integration over the Wishart (or pseudo-Wishart) distribution.…”

Posterior contraction rates for constrained deep Gaussian processes in density estimation and classification

“…Then A and B are convex symmetric subsets of R 2n . Hence, from the Gaussian correlation inequality ( [38], see also [24]), we obtain…”

Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities

“…if B is real, where the expectation refers to the random W m (2α, I m )-Wishart (or pseudo-Wishart) matrix S (see [16] or [18]). If there are one or more pure imaginary columns in B, caused by some negative eigenvalues of…”