Independence preserving property of Kummer laws

Abstract

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\mathit{\alpha },\mathit{\beta }\ge 0$, $\mathit{\alpha }\ne \mathit{\beta }$,

$${\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}(x,y)=\left(y\frac{1+\mathit{\beta }(x+y)}{1+\mathit{\alpha }x+\mathit{\beta }y},x\frac{1+\mathit{\alpha }(x+y)}{1+\mathit{\alpha }x+\mathit{\beta }y}\right),$$

then the random variables U and V defined by $(U,V)={\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}(X,Y)$ 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 ${\mathit{\psi }}_{\mathit{\alpha },\mathit{\beta }}$ 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

$$U=\frac{Y}{1+X}\text{and}V=X\left(1+\frac{Y}{1+X}\right),$$

which corresponds to the case of ${\mathit{\psi }}_{1,0}$. 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.