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 models.

翻译：我们证明：若 $X,Y$ 是正的、独立的、非狄拉克随机变量，且对于 $\alpha,\beta\ge 0$，$\alpha\neq \beta$，有 $$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ 则由 $(U,V)=\psi_{\alpha,\beta}(X,Y)$ 定义的随机变量 $U$ 与 $V$ 相互独立当且仅当 $X$ 和 $Y$ 服从具有适当相关参数的库默尔分布。换言之，在Croydon和Sasada于文献\cite{CS2020}中引入的框架下，由$\psi_{\alpha,\beta}$ 控制的格点递归模型的任何不变测度必为具有库默尔边缘分布的乘积测度。该结果将早期通过独立性 $$ U=\tfrac{Y}{1+X}\quad\text{和}\quad V= X\left(1+\tfrac{Y}{1+X}\right)$$ 刻画库默尔定律和伽马定律的结论（对应于 $\psi_{1,0}$ 情形）进行了推广。我们还证明，库默尔律的这一独立性性质涵盖文献中已知的若干独立性模型作为极限情形：卢卡奇模型、库默尔-伽马模型、松本-约尔模型以及离散科特韦赫-德弗里斯模型。