Integral representations and approximations for multivariate gamma distributions

Let R be a p x p-correlation matrix with an "m-factorial" inverse R-1 = D-BB' with diagonal D minimizing the rank m of B. A new (2(m+1))-variate integral representation is given for p-variate gamma distributions belonging to R, which is based on the above decomposition of R-1 without the restriction D > 0 required in former formulas. This extends the applicability of formulas with small T. For example, every p-variate gamma cdf can be computed by an at most (2(p-1))-variate integral if p = 3 or p = 4. Since computation is only feasible for small m, a given R is approximated by an m-factorial R-0. The cdf belonging to R is approximated by the cdf associated with R-0 and some additional correction terms with the deviations between R and R-0.