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Limit theorems for the fractional nonhomogeneous Poisson process

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  12 July 2019

Nikolai Leonenko ,
Enrico Scalas  and
Mailan Trinh
Nikolai Leonenko*
Affiliation:
Cardiff University
Enrico Scalas*
Affiliation:
University of Sussex
Mailan Trinh*
Affiliation:
University of Sussex
*
*Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
**Postal address: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
**Postal address: Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, UK.
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Abstract

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Keywords

Fractional point processeslimit theoremPoisson processadditive processLévy processtime changesubordination

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 246 - 264
Copyright
© Applied Probability Trust 2019 

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