  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.
