A martingale characterization of mixed Poisson processes

1987 ◽  
Vol 24 (1) ◽  
pp. 246-251 ◽  
Author(s):  
Dietmar Pfeifer ◽  
Ursula Heller
Keyword(s):  
Poisson Process ◽  
Random Variable ◽  
Poisson Processes ◽  
Birth Process ◽  
Mixed Poisson Process ◽  
Martingale Characterization

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

A martingale characterization of mixed Poisson processes

1987 ◽  
Vol 24 (01) ◽  
pp. 246-251 ◽  
Author(s):  
Dietmar Pfeifer ◽  
Ursula Heller
Keyword(s):  
Poisson Process ◽  
Random Variable ◽  
Poisson Processes ◽  
Birth Process ◽  
Mixed Poisson Process ◽  
Martingale Characterization

It is shown that an elementary pure birth process is a mixed Poisson process iff the sequence of post-jump intensities forms a martingale with respect to the σ -fields generated by the jump times of the process. In this case, the post-jump intensities converge almost surely to the mixing random variable of the process.

scholarly journals On the equivalence of various definitions of mixed poisson processes

2019 ◽  
Vol 69 (2) ◽  
pp. 453-468
Author(s):  
Demetrios P. Lyberopoulos ◽  
Nikolaos D. Macheras ◽  
Spyridon M. Tzaninis
Keyword(s):  
Probability Distribution ◽  
Poisson Process ◽  
Random Variable ◽  
Poisson Processes ◽  
Counting Processes ◽  
Mixing Parameter ◽  
Mixed Poisson Process ◽  
Probability Spaces ◽  
The One

Abstract Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing probability 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 for the characterization of mixed Poisson processes in terms of disintegrations cannot be omitted.

scholarly journals A martingale characterization of Pólya-Lundberg processes

2006 ◽  
Vol 43 (03) ◽  
pp. 741-754 ◽  
Author(s):  
Birgit Niese
Keyword(s):  
Poisson Process ◽  
Exponential Family ◽  
Exponential Families ◽  
Counting Processes ◽  
Mixed Poisson Process ◽  
Martingale Characterization

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

scholarly journals A martingale characterization of Pólya-Lundberg processes

2006 ◽  
Vol 43 (3) ◽  
pp. 741-754 ◽  
Author(s):  
Birgit Niese
Keyword(s):  
Poisson Process ◽  
Exponential Family ◽  
Exponential Families ◽  
Counting Processes ◽  
Mixed Poisson Process ◽  
Martingale Characterization

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

An extension of watanabe's theorem of characterization of poisson processes over the positive real half line

1975 ◽  
Vol 12 (2) ◽  
pp. 396-399 ◽  
Author(s):  
P. Bremaud
Keyword(s):  
Poisson Process ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Characterization Theorem ◽  
Poisson Processes ◽  
Half Line ◽  
Positive Real ◽  
The Mean ◽  
Martingale Characterization

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

An extension of watanabe's theorem of characterization of poisson processes over the positive real half line

1975 ◽  
Vol 12 (02) ◽  
pp. 396-399 ◽  
Author(s):  
P. Bremaud
Keyword(s):  
Poisson Process ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Characterization Theorem ◽  
Poisson Processes ◽  
Half Line ◽  
Positive Real ◽  
The Mean ◽  
Martingale Characterization

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

scholarly journals Some characterizations for Markov processes as mixed renewal processes

2018 ◽  
Vol 68 (6) ◽  
pp. 1477-1494 ◽  
Author(s):  
Nikolaos D. Macheras ◽  
Spyridon M. Tzaninis
Keyword(s):  
Markov Processes ◽  
Basic Problem ◽  
Markov Property ◽  
Random Variable ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Mixing Parameter ◽  
Mild Assumption ◽  
Mixed Poisson Process ◽  
Original Class

Abstract In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector defined by Lyberopoulos and Macheras (enlarging Huang’s original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples, we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.

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