# Integral representations for convolutions of non-central multivariate gamma distributions

## TL;DR

This paper presents new integral representations for the cumulative distribution functions of convolutions of non-central multivariate gamma distributions, facilitating more efficient numerical computation and analysis.

## Contribution

It introduces three types of integral formulas for these distributions, including joint distributions of quadratic forms, improving computational efficiency over traditional methods.

## Key findings

- Derived integral representations for convolutions of non-central multivariate gamma distributions.
- Obtained joint distribution of diagonal elements of quadratic forms involving normal vectors.
- Integral formulas are numerically more favorable than Fourier or Laplace inversion methods.

## Abstract

Three types of integral representations for the cumulative distribution functions of convolutions of non-central p-variate gamma distributions are given by integration of elementary complex functions over the p-cube Cp = (-pi,pi]x...x(-pi,pi]. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX' with n independent normally distributed column vectors in X is obtained. For a single p-variate gamma distribution function (p-1)-variate integrals over Cp-1 are derived. The integrals are numerically more favourable than integrals obtained from the Fourier or laplace inversion formula.