Abstract
The Dufresne laws are defined on the positive line by their Mellin transform \(s \mapsto (a_1 )_s \cdots (a_p )_s /(b_1 )_s \cdots (b_q )_s = ({\text{a)}}_s /(b)_s \), where the a i and b j are positive numbers, with p≥q, and where (x) s denotes Γ(x+s)/Γ(x). Typical examples are the laws of products of independent random variables with gamma and beta distributions. They occur as the stationary distribution of certain Markov chains (X n) on \(\mathbb{R}\) defined by \(X_n = A_n (X_{n - 1} + B_n )\) where X 0, (A 1, B 1),..., (A n, B n),... are independent. This paper gives some explicit examples of such Markov chains. One of them is surprisingly related to the golden number. While the properties of the product of two independent Dufresne random variables are trivial, we give several properties of their sum: the hypergeometric functions are the main tool here. The paper ends with an extension of these Dufresne laws to the space of positive definite matrices and to symmetric cones.
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Chamayou, JF., Letac, G. Additive Properties of the Dufresne Laws and Their Multivariate Extension. Journal of Theoretical Probability 12, 1045–1066 (1999). https://doi.org/10.1023/A:1021649305082
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DOI: https://doi.org/10.1023/A:1021649305082