On the equivalence of various definitions of mixed Poisson processes

2016/07/19 by Demetrios P. Lyberopoulos, Lyberopoulos, D. P., N. D. Macheras +3
Economics, Econometrics and Finance · Mathematics · #28A50 #60G05 #60J27 #60K05 #91B30 #Advanced Banach Space Theory #FOS: Mathematics #Point processes and geometric inequalities #Primary 60G55 #Probability (math.PR) #Secondary 60A10 #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1607.05452

openalex publication_date 2016/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization of each one of the above mixed Poisson processes in terms of disintegrations is provided. Moreover, some examples of "canonical" probability spaces admitting counting processes satisfying the equivalence of all above statements are given. Finally, it is shown that our assumptions are essential for the characterization of mixed Poisson processes in terms of disintegrations.

Citations

• Some characterizations for Markov processes as mixed renewal processes

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