Superposition of time-changed Poisson processes and their hitting times

TL;DR

This paper investigates extensions of Poisson processes of order with time changes via Bernstein subordinators, analyzing their hitting times and distributional properties, including cases where processes can skip states with positive probability.

Contribution

It introduces new models of time-changed Poisson processes with varied weights and analyzes their hitting times, providing explicit distributions and insights into their jump behaviors.

Findings

Explicit distributions for hitting times of extended processes

Processes can skip states with positive probability

Multiple upward jumps influence state visitation

Abstract

The Poisson process of order $i$ is a weighted sum of independent Poisson processes and is used to model the flow of clients in different services. In the paper below we study some extensions of this process, for different forms of the weights and also with the time-changed versions, with Bern\v stein subordinator playing the role of time. We focus on the analysis of hitting times of these processes obtaining sometimes explicit distributions. Since all the processes examined display a similar structure with multiple upward jumps sometimes they can skip all states with positive probability even on infinitely long time span.