正稳定随机变量幂的无限可分性
Infinite divisibility of powers of positive stable random variables
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- School of Mathematics and Statistics, Wuhan University of Technology（武汉理工大学数学与统计学院）
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中文总结 AI 辅助
本文证明正$\alpha$-稳定随机变量$Z_\alpha$的任意正幂$Z_\alpha^p$无限可分，并结合负幂刻画得到$p\ne0$时无限可分的充要条件，证明基于Kanter分解与乘积性质。
中文摘要 AI 辅助
设$Z_\alpha$为一个正的$\alpha$-稳定随机变量，其中$0<\alpha<1$。我们证明对于每一个$p>0$，$Z_\alpha^p$是无限可分的。将此结果与Jedidi和Simon关于$Z_\alpha$负幂无限可分性的刻画相结合，我们得到，对于$p\ne0$，$$Z_\alpha^p\\ \text{是无限可分的} \quad\Longleftrightarrow\quad p\in\left(-\infty,-\frac{\alpha}{1-\alpha}\right]\cup(0,\infty).$$ 证明依赖于Kanter分解和一个一般的乘积性质：若$X$是指数分布的混合，$W$是广义伽马卷积，且$X$与$W$独立，则对于每个$c\ge 0$，$(X+c)W$是无限可分的。
英文摘要
Let $Z_α$ be a positive $α$-stable random variable, where $0<α<1$. We prove that $Z_α^p$ is infinitely divisible for every $p>0$. Combining this result with Jedidi and Simon's characterization of infinite divisibility of negative powers of $Z_α$, we obtain, for $p\ne0$, $$Z_α^p\ \text{is infinitely divisible} \quad\Longleftrightarrow\quad p\in\left(-\infty,-\fracα{1-α}\right]\cup(0,\infty).$$ The proof relies on Kanter's factorization and a general product property: if $X$ is a mixture of exponential distributions, $W$ is a generalized gamma convolution, and $X$ and $W$ are independent, then $(X+c)W$ is infinitely divisible for every $c\ge 0$.
