发表机构
School of Mathematics and Statistics, Wuhan University of Technology; Institute for Advanced Study in Mathematics, Harbin Institute of Technology（武汉理工大学数学与统计学院; 哈尔滨工业大学数学前沿研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了$\alpha$-柯西分布自可分解当且仅当$\alpha=2$，基于对称情形的新判据，并应用于对称稳定过程击中时间，表明指数$1<\alpha<2$时击中时间不自可分解。
AI 中文摘要
在2009年，Yano、Yano和Yor在研究对称稳定过程的击中时间时，提出了$\alpha$-柯西分布是否自可分解或至少无限可除的问题。最近，第一作者证明了$\alpha$-柯西分布是无限可除的当且仅当$1 < \alpha \leq 2$。在本文中，我们证明了$\alpha$-柯西分布是自可分解的当且仅当$\alpha = 2$。该证明基于对称情形下自可分解性的一个新判据。作为应用，我们证明了从零出发的指数为$1<\alpha<2$的对称稳定过程首次击中非零点的时刻不是自可分解的。
英文摘要
In 2009, Yano, Yano and Yor asked whether the $α$-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that $α$-Cauchy distribution is infinitely divisible if and only if $1 < α\leq 2$. In this paper, we prove that $α$-Cauchy distribution is self-decomposable if and only if $α= 2$. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index $1<α<2$ starting from zero is not self-decomposable.
Comments9 pages
