On the infinite divisibility of distributions of some inverse subordinators

TL;DR

This paper investigates whether distributions of certain inverse subordinators are infinitely divisible, showing many are not, including the fractional Poisson process, using tail probability bounds.

Contribution

It provides new results demonstrating non-infinite divisibility of distributions of well-known inverse subordinators and related processes.

Findings

Distributions of many inverse subordinators are not infinitely divisible.

The distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible.

The fractional Poisson process distribution is not infinitely divisible.

Abstract

We consider the infinite divisibility of distributions of some well-known inverse subordinators. Using a tail probability bound, we establish that distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of the fractional Poisson process is not infinitely divisible.