Some probabilistic properties of fractional point processes

TL;DR

This paper investigates the probabilistic properties of fractional point processes, including hitting times and renewal characteristics, with explicit results for space-fractional Poisson processes and extensions involving generalized grey Brownian motions.

Contribution

It provides explicit hitting probabilities for space-fractional Poisson processes and analyzes extended processes involving sums of subordinators and generalized grey Brownian motions.

Findings

Explicit hitting probabilities for space-fractional Poisson processes.

The space-time Poisson process is not a renewal process.

Probabilistic features of extended counting processes are characterized.

Abstract

This paper studies the first hitting times of generalized Poisson processes $N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson processes, $N^\alpha(t)$, $t>0$ (corresponding to $f= x^\alpha$), the hitting probabilities $P\{T_k^\alpha<\infty\}$ are explicitly obtained and analyzed. The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form $N(|\mathcal{G}_{H,\nu}(t)|)$ where $\mathcal{G}_{H,\nu}(t)$ are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.