  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.
