On Distributions of Certain State Dependent Fractional Point Processes

TL;DR

This paper derives explicit state probability distributions for certain state-dependent fractional point processes using the Adomian decomposition method, simplifying the inversion of Laplace transforms and exploring related Mittag-Leffler distributions.

Contribution

It introduces a novel application of the Adomian decomposition method to obtain explicit solutions for state-dependent fractional point processes, improving upon cumbersome Laplace transform inversions.

Findings

Explicit state probabilities derived for various processes

Distribution formulas for convolutions of Mittag-Leffler variables

Simplified approach to solving difference differential equations

Abstract

We obtain the explicit expressions for the state probabilities of various state dependent fractional point processes recently introduced and studied by Garra et al. (2015). The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference differential equations governing the state probabilities of these state dependent processes. The distributions of some convolutions of the Mittag-Leffler random variables are derived as special cases of the obtained results.