A Random Difference Equation with Dufresne Variables revisited

Abstract

The Dufresne laws (laws of product of independent random variables with gamma and beta distributions) occur as stationary distribution of certain Markov chains $X_n$ on $R$ defined by: \begin{equation} X_n = A_n ( X_{n-1} + B_n ) \end{equation} where $X_0,(A_1,B_1),...,(A_n,B_n)$ are independent and the $(A_i,B_i{)}^{\prime }$s are identically distributed. This paper generalizes an explicit example where $A$ is the product of two independent ${\beta }_{a,1},{\beta }_{b,1}$ and $B\sim {\gamma }_1$ or ${\gamma }_2$. Keywords: beta, gamma and Dufresne distributions,Markov chains, stationary distributions, hypergeometric differential equations, Poisson process.

arXiv comment11 pages, 2 tables, 1 figure