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.

Infinitely divisible (ID) distributions were introduced by de Finetti in 1929. Ever since the research literature on these distributions is growing rapidly. A real-valued random variable

In recent years time-changed stochastic processes are getting increased attention due to their applications in finance, geophysics, fractional partial differential equations and in modeling the anomalous diffusion in statistical physics (see Janczura et al., [

In this article we study the infinite divisibility of the distribution of some inverse subordinators corresponding to drift-less subordinators. We first obtain a bound on the tail probability of these inverse subordinators. We establish that the distributions of inverse stable, inverse tempered stable and first-exit times of inverse Gaussian subordinators are not ID. Further, we also show that the distribution of a renewal process time-changed by ISS is not ID. In particular we establish that the distribution of the fractional Poisson process is not ID.

One should not conclude from these results that the distributions of inverse subordinators are not ID in general. One counter-example is the Poisson process. Let

Further, the fractional Poisson process, for which applications are suggested in insurance (Biard and Saussereau, [

ID distributions are at the heart of the theory of Lévy processes. Every continuous-time Lévy process has distributions that are necessarily ID (see, e.g., Steutel and Van Harn, [

A subordinator is a one-dimensional Lévy process that is non-decreasing almost sure (a.s.). Such processes can be thought of as a random model of time evolution. If

To prove the non-infinite divisibility of inverse subordinators in this article, we use the tail bound (

For an

It is worthwhile to mention the results about

Next we prove the non-infinite divisibility of distributions of inverse tempered stable subordinators (ITSS). Tempered stable subordinators (TSS) are obtained by exponential tempering in distributions of stable subordinators (see, e.g., Rosiński, [

The Laplace exponent for ITSS is given by

Next we discuss the non-infinite divisibility of the distribution of inverse of an inverse Gaussian subordinator. It is worth to mention that an inverse Gaussian subordinator is a particular case of TSS. Let

Note that when

A proof of non-infinite divisibility of distribution of

Next, we discuss the tail probabilities for gamma subordinators. Let

Next we discuss some transformed processes of the inverse subordinators. Consider the transformed ISS

We here provide the proof for an inverse gamma subordinator only. Proofs for other subordinators follow similarly. Note that

We can easily show that

For a strictly increasing subordinator

Nane [

By self-similarity of

Let

Note that

Meerschaert et al. [