On Infinite Divisibility of the Distribution of Some Inverse Subordinators

We consider the infinite divisibility of the distributions of some well known inverse subordinators. Using a tail probability bound, we establish that the 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 fractional Poisson process is not infinitely divisible.