Superposition of time-changed Poisson processes and their hitting times

2019/09/29 by Aditya Maheshwari, Enzo Orsingher, Maheshwari, A. +3
Business, Management and Accounting · Computer Science · Economics, Econometrics and Finance · #60G51 #60G55 #Advanced Queuing Theory Analysis #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1909.13213

openalex publication_date 2019/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory
• Fractional Poisson Process
• The space-fractional Poisson process
• Counting processes with Bernštein intertimes and random jumps
• On the long-range dependence of fractional Poisson and negative binomial processes

Related

• Time-changed Poisson processes of order k
• First Steps Towards an Imprecise Poisson Process
• Poisson Arrivals See Time Averages
• Geometrical subordinated Poisson processes and its extensions
• Compound Poisson Process with a Poisson Subordinator