P-Thinned Gamma Process and Corresponding Random Walk

P-thinned Gamma processes could be considered as a particular case of renewal processes which inter-renewal times are zero-inflated Gamma distributed. This paper considers also the difference between two, not obligatory identically distributed, processes which time intersections coincide in distribution with convolutions of zero-inflated Gamma distributed random variables. The idea comes from the Variance-Gamma model which is defined as Gamma time changed Wiener processes and is stochastically equivalent to a difference between two independent Gamma processes. The main properties and numerical characteristics of the resulting process are obtained. Simulation illustrates the theoretical results.