Limit Theorems for the Fractional Non-homogeneous Poisson Process

TL;DR

This paper develops limit theorems for fractional non-homogeneous Poisson processes and their compound variants, using martingale methods and regular variation, with verification through simulations.

Contribution

It introduces a fractional non-homogeneous compound Poisson process and establishes new finite-dimensional and functional limit theorems for these processes.

Findings

Finite-dimensional limit theorems derived

Functional limit theorems established

Simulation verifies theoretical results

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.