Gamma分布和Kummer分布刻画中测度技术的变化
Change of measure technique in characterizations of the Gamma and Kummer distributions
摘要：
如果$X$和$Y$是具有分布$μ$和$ν$的独立随机变量，则对于某些$ψ$和$ϕ$，$U=ψ(X,Y)$和$V=ϕ(X,Y)$也是独立的。这种类型的属性是已知的许多重要的概率分布$μ$和$ν$。也已经广泛研究了相关的表征问题：设$X$和$Y$是独立的，且设$U$和$V$是独立的。$X$和$Y$的分布是否分别为$μ$和$ν$？最近，文献中出现了两个新的性质和这类涉及库默尔分布的特征。对于具有伽马和库默分布的独立$X$和$Y$，Koudou和Vallois观察到$U=(1+(X+Y)^{-1})/(1+X^{-1})$和$V=X+Y$也是独立的，Hamza和Vallois观察到$U=Y/(1+X)$和$V=X(1+Y/(1+X))$是独立的。在2011年和2012年，Koudou，Vallois证明了与第一个属性相关的特征，而第二个设置中的特征最近在Piliszek，Wesoenkowski（2016）中给出。在这两种情况下，都需要对$X$和$Y$密度的光滑性进行技术假设。2015年，在第一种设置中$U$和$V$独立的假设被削弱为$U$和$U^{-1}$回归的恒定性，给出了没有密度假设的$V$。然而，引入了附加假设$\mathbb{E} X^{-1}<\infty$。在本文中，我们提供了一个完整的答案，在这两种设置的表征问题，没有任何额外的技术假设，关于平滑度或存在的时刻。该方法是，首先，通过利用一些条件的$U$回归$V$，这是弱于独立，但目前的假设是必要的特征。第二，使用的技术变化的措施，我们表明，时刻的假设可以避免。
英文摘要：
If $X$ and $Y$ are independent random variables with distributions $μ$ and $ν$ then $U=ψ(X,Y)$ and $V=ϕ(X,Y)$ are also independent for some $ψ$ and $ϕ$. Properties of this type are known for many important probability distributions $μ$ and $ν$. Also related characterization questions have been widely investigated: Let $X$ and $Y$ be independent and let $U$ and $V$ be independent. Are the distributions of $X$ and $Y$ $μ$ and $ν$, respectively? Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent $X$ and $Y$ with gamma and Kummer distributions Koudou and Vallois observed that $U=(1+(X+Y)^{-1})/(1+X^{-1})$ and $V=X+Y$ are also independent, and Hamza and Vallois observed that $U=Y/(1+X)$ and $V=X(1+Y/(1+X))$ are independent. In 2011 and 2012 Koudou, Vallois characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in Piliszek, Wesołowski (2016). In both cases technical assumptions on smoothness properties of densities of $X$ and $Y$ were needed. In 2015, the assumption of independence of $U$ and $V$ in the first setting was weakened to constancy of regressions of $U$ and $U^{-1}$ given $V$ with no density assumptions. However, the additional assumption $\mathbb{E} X^{-1}<\infty$ was introduced. In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of $U$ given $V$, which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided.
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