We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha,\beta\ge 0$, $\alpha\neq \beta$, $$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=\psi_{\alpha,\beta}(X,Y)$ 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 $\psi_{\alpha,\beta}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020}, is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $\psi_{1,0}$. We also show 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 ones.

翻译：我们证明：如果$X,Y$是正独立非狄拉克随机变量，且对于$\alpha,\beta\ge 0$、$\alpha\neq \beta$，满足$$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$则通过$(U,V)=\psi_{\alpha,\beta}(X,Y)$定义的随机变量$U$和$V$相互独立当且仅当$X$和$Y$服从参数适当相关的Kummer分布。换言之，Croydon与Sasada在文献\cite{CS2020}提出的框架中，由$\psi_{\alpha,\beta}$驱动的任何格点递归模型的不变测度必然是具有Kummer边际分布的乘积测度。该结果推广了早期通过下列独立性对Kummer分布和Gamma分布的特征刻画：$$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ 这对应于$\psi_{1,0}$的情形。我们还证明，Kummer分布的此独立性性质在极限情形下涵盖了文献中已知的若干独立性模型：Lukacs模型、Kummer-Gamma模型、Matsumoto-Yor模型以及离散Korteweg de Vries模型。