Independence preserving property of Kummer laws

We prove that if X,Y are positive, independent, non-Dirac random variables and if for α,β≥0, α≠β, ψα,β(x,y)=y1+β(x+y)1+αx+βy,x1+α(x+y)1+αx+βy, then the random variables U and V defined by (U,V)=ψα,β(X,Y) are independent if and only if X and Y follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by ψα,β in the scheme introduced by Croydon and Sasada in (Croydon and Sasada (2020)) is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of U=Y1+XandV=X1+Y1+X, which corresponds to the case of ψ1,0. We also show, in the supplement, that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.