Occupation times of alternating renewal processes with Lévy applications

Abstract

This paper presents a set of results relating to the occupation time $α(t)$ of a process $X(\cdot)$. The first set of results concerns exact characterizations of $α(t)$ for $t\geq0$, e.g., in terms of its transform up to an exponentially distributed epoch. In addition we establish a central limit theorem (entailing that a centered and normalized version of $α(t)/t$ converges to a zero-mean Normal random variable as $t\rightarrow\infty$) and the tail asymptotics of $P(α(t)/t\geq q)$. We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.