Geometrical subordinated Poisson processes and its extensions

TL;DR

This paper introduces a new class of Poisson processes subordinated by geometric counting processes, extending existing models and analyzing their properties for applications like shock modeling.

Contribution

It defines and studies the properties of geometric subordinated Poisson processes and their variants, extending prior work with new distributional and asymptotic analyses.

Findings

Derived distributional properties of the new processes.

Analyzed asymptotic correlation behavior.

Discussed applications in shock modeling.

Abstract

In this paper, we study a generalized version of the Poisson-type process by time-changing it with the geometric counting process. Our work generalizes the work done by Meoli (2023) \cite{meoli2023some}. We defined the geometric subordinated Poisson process (GSPP), the geometric subordinated compound Poisson process (GSCPP) and the geometric subordinated multiplicative Poisson process (GSMPP) by time-changing the subordinated Poisson process, subordinated compound Poisson process and subordinated multiplicative Poisson process with the geometric counting process, respectively. We derived several distributional properties and many special cases from the above-mentioned processes. We calculate the asymptotic behavior of the correlation structure. We have discussed applications of time-changed generalized compound Poisson in shock modelling.