A Random Difference Equation with Dufresne Variables revisited

TL;DR

This paper revisits a class of Markov chains with stationary distributions related to Dufresne laws, generalizing explicit examples involving beta and gamma distributions, and explores their mathematical properties.

Contribution

It extends previous work by providing a generalized explicit example of the chain where the multiplicative factor is a product of beta distributions and the additive component is gamma-distributed.

Findings

Identifies new explicit solutions for the stationary distribution

Links Dufresne laws to hypergeometric differential equations

Provides insights into the structure of Markov chains with these distributions

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 $(A_i,B_i)'$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.