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.

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.

Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation

1995 ◽  
Vol 27 (02) ◽  
pp. 367-383 ◽  
Author(s):  
Jesper Møller
Keyword(s):  
Comparative Analysis ◽  
Poisson Processes ◽  
Three Dimensions ◽  
Empirical Results ◽  
Random Nucleation ◽  
Inhomogeneous Poisson Processes

Simulation procedures for typical Johnson-Mehl crystals generated under various models for random nucleation are proposed. These procedures include algorithms for simulating spatio-time-inhomogeneous Poisson processes. Empirical results for a particular class of Johnson-Mehl tessellations in two and three dimensions show remarkably different crystals.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (03) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

ON THE IDENTIFICATION OF THE SOURCE OF EMISSION ON THE PLANE

2019 ◽  
Vol 53 (2 (249)) ◽  
pp. 75-81
Author(s):  
N.V. Arakelyan ◽  
Yu.A. Kutoyants
Keyword(s):  
Asymptotic Properties ◽  
Radioactive Source ◽  
Poisson Processes ◽  
Inhomogeneous Poisson Processes ◽  
The Moment ◽  
Source Emission

We consider the problem of identification of the position and the moment of the beginning of a radioactive source emission on the plane. The acts of emission constitute inhomogeneous Poisson processes and are registered by $ K $ detectors on the plane. We suppose that the moments of arriving of the signals at the detectors are measured with some small errors. Then, using these estimate, we construct the estimators of the position of source and the moment of the beginning of emission. We study the asymptotic properties of these estimators for large signals and prove their consistency.

scholarly journals A statistical model investigating the prevalence of tuberculosis in New York City using counting processes with two change-points

2008 ◽  
Vol 136 (12) ◽  
pp. 1599-1605 ◽  
Author(s):  
J. A. ACHCAR ◽  
E. Z. MARTINEZ ◽  
A. RUFFINO-NETTO ◽  
C. D. PAULINO ◽  
P. SOARES
Keyword(s):  
New York ◽  
New York City ◽  
Monte Carlo ◽  
Markov Chain ◽  
Bayesian Analysis ◽  
York City ◽  
Poisson Processes ◽  
Change Points ◽  
Counting Processes ◽  
Parameters Of Interest

SUMMARYWe considered a Bayesian analysis for the prevalence of tuberculosis cases in New York City from 1970 to 2000. This counting dataset presented two change-points during this period. We modelled this counting dataset considering non-homogeneous Poisson processes in the presence of the two-change points. A Bayesian analysis for the data is considered using Markov chain Monte Carlo methods. Simulated Gibbs samples for the parameters of interest were obtained using WinBugs software.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (3) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

Sign in / Sign up

Export Citation Format

Share Document