Independences of Kummer laws

12/06/2022

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by   Efoevi Angelo Koudou, et al.

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We prove that if X, Y are positive, independent, non-Dirac random variables and if α, β ≥ 0, α = β, then the random variables U and V defined by U = Y 1+β(X+Y) 1+αX+βY and V = X 1+α(X+Y) 1+αX+βY are independent if and only if X and Y follow Kummer distributions with suitable parameters. In other words, the Kummer distributions are the only invariant measures for lattice recursion models introduced by Croydon and Sasada in [3]. The result extends earlier characterizations of Kummer and gamma laws by independence of U = Y 1+X and V = X 1 + Y 1+X , which is the case of (α, β) = (1, 0).