# Limit theorems for the fractional non-homogeneous Poisson process

Leonenko, Nikolai, Scalas, Enrico and Trinh, Mailan (2019) Limit theorems for the fractional non-homogeneous Poisson process. Journal of Applied Probability. ISSN 0021-9002 (Accepted)

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## Abstract

The fractional non-homogeneous Poisson process was introduced by a time change of the non-homogeneous Poisson process with the inverse $\alpha$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous 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.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Probability and Statistics Research Group Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics > QA0274 Stochastic processesQ Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics Organisation Enrico Scalas 31 Jan 2019 11:48 31 Jan 2019 11:50 http://srodev.sussex.ac.uk/id/eprint/81576