Geometrical subordinated Poisson processes and its extensions
Neha Gupta, Aditya Maheshwari, Dheeraj Goyal
TL;DR
The paper develops a unified framework for generalized Poisson-type processes time-changed by a geometric counting process, introducing the geometric subordinated Poisson process (GSPP) and its compound and multiplicative extensions (GSCPP, GSMPP). It derives explicit distributional forms, moment relations, first-passage time results, and long-range dependence properties, highlighting special cases such as SFPP and TSFPP. The authors then apply these models to shock-based reliability problems, obtaining closed-form reliability and hazard expressions for extreme and cumulative shock scenarios under geometric time-change. Overall, the work offers a flexible, analytically tractable toolkit for modeling clustered, heavy-tailed event arrivals in reliability, finance, and related shock-model contexts.
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.