Limit theorems for the fractional nonhomogeneous Poisson process

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse $\alpha$-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.