论文标题
独立保存库默法律的财产
Independence preserving property of Kummer laws
论文作者
论文摘要
我们证明，如果$ x，y $是正面的，独立的，非迪拉克随机变量，如果以$α，β\ ge 0 $，$α\ neqβ$，$ $ ψ_{α,β}(x,y)=\left(y\,\tfrac{1+β(x+y)}{1+αx+βy},\;x\,\tfrac{1+α(x+y)}{1+αx+βy}\right), $$ then the random variables $U$ and $V$ defined当$（u，v）=ψ_{α，β}（x，y）$是独立的，并且仅当$ x $和$ y $遵循具有适当相关参数的kummer发行版时。换句话说，由Croydon和Sasada在\ cite {CS2020}中引入的方案中，由$ψ_{α，β} $控制的晶格递归模型的任何不变度量必然是具有Kummer Marginals的产品度量。该结果通过$$ U = \ tfrac {y} {1+x} {1+x} \ quad \ mbox {and} \ quad v = x \ left（1+ \ tfrac {y} {y} {y} {1+x} {1+x} {1+x} {1+x} {1+x} {1+x} {1+x} {1+x}，$ consects $ contry的独立性扩展了Kummer和Gamma定律的早期特征。 我们还表明，Kummer法律的独立性涵盖了文献中已知的几种独立模型：Lukacs，Kummer-Gamma，Matsumoto-Yor和离散的Korteweg de Vries模型。
We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $α,β\ge 0$, $α\neq β$, $$ ψ_{α,β}(x,y)=\left(y\,\tfrac{1+β(x+y)}{1+αx+βy},\;x\,\tfrac{1+α(x+y)}{1+αx+βy}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=ψ_{α,β}(X,Y)$ 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 $ψ_{α,β}$ 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 $ψ_{1,0}$. 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.
