Integral Representations for Convolutions of Non–central Multivariate Gamma Distributions

Three types of integral representations for the cumulative distribution functions of convolutions of Γp(αk,Σk,∆k)–distributions with non–centrality matrices ∆k are given by integration of products of simple complex functions over the p–cube (−π, π]. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX ′ with n independentNp(μk,Σ)–distributed columns in Xp×n and a fixed A ≥ 0 is obtained. For a single Γp(α,Σ,∆)–cdf (p− 1)–variate integrals over (−π, π] are derived. The integrals are numerically more favourable than integrals obtained from the Fourier– or Laplace inversion formula.