Stationary Regenerative Sets and Subordinators

Abstract

In this paper we give a simple construction of the general stationary regenerative set, based on the stationary version of the associated subordinator (increasing Levy process). We show that, in a certain sense, the closed range of such a Levy process is a stationary regenerative subset of $\mathbb{R}$. The distribution of this regenerative set is $\sigma$-finite in general; it is finite $\operatorname{iff}$ the increments of the Levy process have finite expectation.