Abstract

We obtain the distribution of the sum of independent and non-identically distributed generalized Mittag–Leffler random variables. We then apply this result to study some related fractional point processes. We present their explicit probability mass functions as well as their connections with the fractional integral/differential equations. In the case of a point process with Mittag–Leffler distributed waiting times which alternate two indexes ${\nu }_1,{\nu }_2\in (0,1]$ and two rates ${\lambda }_1,{\lambda }_2>0,$ we also study the conditional arrival times and we show an application to the telegraph process.

中文翻译：

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关于独立广义 Mittag-Leffler 随机变量和相关分数过程的总和

摘要

我们获得了独立和不同分布的广义 Mittag-Leffler 随机变量之和的分布。然后我们应用这个结果来研究一些相关的分数点过程。我们展示了它们的显式概率质量函数以及它们与分数积分/微分方程的联系。在具有 Mittag-Leffler 分布式等待时间的点过程的情况下，交替两个索引${\nu }_1,{\nu }_2\in (0,1]$ 和两个费率 ${\lambda }_1,{\lambda }_2>0,$ 我们还研究了有条件的到达时间，并展示了对电报过程的应用。