A multivariate gamma distribution arising from a Markov model

A Markov chain{Xt}, which has been useful for modelling in hydrology, can be specified by the Laplace transform (LT) of the conditional p.d.f. ofXt+1 givenXt=xt, which is assumed to be of the ‘exponential’ formH(θ)exp{-G(θ)xt}. For appropriate choice ofH andG the marginal distribution ofXt is the (univariate) gamma distribution. In this case, the joint p.d.f. ofXt+1,...,Xt+n and its LT, are obtained, and this is extended to a seasonal version of the chain. A simple method of generating observations from these multivariate gamma distributions is noted, and the joint LT is applied to the problem of determining moments of weighted sums of such variables.