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Limit theorems for the fractional nonhomogeneous Poisson process

Published online by Cambridge University Press:  12 July 2019

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.

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.

Type
Research Papers
Copyright
© Applied Probability Trust 2019 

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