Additive properties of the Dufresne laws and their multivariate extension
The Dufresne laws are defined on the positive line by their Mellin transforms. This is a class of distributions parameterized by positive real vectors \((a_1,\dots,a_p)\) and \((b_1,\dots , b_q)\). Special cases are the beta and gamma distributions. Other examples arise as stationary distributions of Markov chains defined by \(X_n=A_n(X_{n-1} + B_n)\). The product of two independent Dufresne-distributed random variables is also Dufresne, with parameters being the concatenation of the original parameters. The additive convolutions are more complicated. Much of the paper is devoted to examining the additive convolution properties of Dufresne laws in ``a somewhat rhapsodic way''. Identities are derived relating moments of sums of certain pairs of independent Dufresne-distributed random variables to hypergeometric functions, from which the equality of some convolutions of Dufresne laws with other Dufresne laws follows, not to mention an intricate identity involving the 3F2 hypergeometric function and the golden mean. The last section defines multivariate Dufresne laws on the cone of positive-definite symmetric \(r\times r\) matrices, which generalize the classical Wishart and beta-Wishart distributions. The main result is that, under fairly general conditions, the multivariable Dufresne laws satisfy the concatenation rule for multiplicative convolutions.
scientific article; zbMATH DE number 1418051