In this paper, motivated by the risk process, we introduce and study a new point process called geometric Pólya-Aeppli process (GPAP), with underlying exponential distribution. We give two equivalent definitions of the process and discuss some of its properties, such as the distribution of the number of GPAP events up to time t, the distribution of the waiting times, etc. The new process is an extension of the well-known Pólya-Aeppli process, as well as the standard geometric process with underlying exponential distribution.

中文翻译：

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几何Pólya-Aeppli过程

在本文中，受风险过程的启发，我们介绍并研究了一种新的点过程，称为几何Pólya-Aeppli过程（GPAP），具有潜在的指数分布。我们对该过程给出两个等效的定义，并讨论其某些属性，例如直到时间t为止的GPAP事件数的分布，等待时间的分布等。新过程是众所周知的过程的扩展Pólya-Aeppli过程以及具有基本指数分布的标准几何过程。