We prove that if X, Y are positive, independent, non-Dirac random variables and if $\alpha$, $\beta$ $\ge$ 0, $\alpha$ = $\beta$, then the random variables U and V defined by U = Y 1+$\beta$(X+Y) 1+$\alpha$X+$\beta$Y and V = X 1+$\alpha$(X+Y) 1+$\alpha$X+$\beta$Y 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 ($\alpha$, $\beta$) = (1, 0).

翻译：我们证明,如果X是正的、独立的、非Dirac随机变量,如果X是正的、独立的、非Difac 随机的变量,如果美元,美元\beta$= ge$0, 美元= alpha$= 美元= 美元= 美元,那么由U= Y 1+美元\ beta$(X+Y) 1+美元= pha$= alpha$= X 1+美元= ALpha$(X+Y) 1+美元= ALpha$+ 美元\ beta$Y,只有X和Y遵循Kummer的分布,并附有适当的参数,这些变量是Croydon和Sasada在[3] 中引入的唯一变异体变体变量U和V= Y 1+X= X 1+Y 1+X,结果扩大了以前对Kummer 和伽马法律的定性,这是美元= (1, 美元\beta$)= 1,0的情况。