Point processes of the Poisson-Skellam family

We study a general non-homogeneous Skellam-type process with jumps of arbitrary fixed sizes.

We express this process in terms of a linear combination of Poisson processes and study several properties, including the summation of independent processes of the same family, some possible decompositions (which present particularly interesting characteristics) and the limit behaviors.

A compound Poisson representation and a discrete approximation are also presented.

Then, we study the fractional integral of the process as well as the iterated integral of the running average.

Finally, we consider some time-changed versions related to Lévy subordinators, connected to the Bernstein functions, and to the inverses of stable subordinators.