An Elementary Approach for Extending Poisson Processes to the Nonstationary Case

1984 ◽  
Vol 91 (10) ◽  
pp. 646 ◽  
Author(s):  
Bruce R. Johnson
Keyword(s):  
Poisson Processes ◽  
Elementary Approach

scholarly journals An elementary derivation of moments of Hawkes processes

2020 ◽  
Vol 52 (1) ◽  
pp. 102-137
Author(s):  
Lirong Cui ◽  
Alan Hawkes ◽  
He Yi
Keyword(s):  
Poisson Processes ◽  
Counting Processes ◽  
Elementary Approach ◽  
Hawkes Processes ◽  
One Dimensional ◽  
Elementary Derivation ◽  
Cox Processes ◽  
Martingale Methods ◽  
Inhomogeneous Poisson Processes ◽  
Detailed Outline

AbstractHawkes processes have been widely used in many areas, but their probability properties can be quite difficult. In this paper an elementary approach is presented to obtain moments of Hawkes processes and/or the intensity of a number of marked Hawkes processes, in which the detailed outline is given step by step; it works not only for all Markovian Hawkes processes but also for some non-Markovian Hawkes processes. The approach is simpler and more convenient than usual methods such as the Dynkin formula and martingale methods. The method is applied to one-dimensional Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, inhomogeneous Poisson processes, and non-Markovian cases. Several results are obtained which may be useful in studying Hawkes processes and other counting processes. Our proposed method is an extension of the Dynkin formula, which is simple and easy to use.

On some characterizations of the poisson process

1989 ◽  
Vol 26 (01) ◽  
pp. 176-181
Author(s):  
Wen-Jang Huang
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Arrival Time ◽  
Random Variables ◽  
Poisson Processes

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

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