Occupation times of alternating renewal processes with L\'evy applications

This paper presents a set of results relating to the occupation time $\alpha (t)$ of a process $X(\cdot )$. The first set of results concerns exact characterizations of $\alpha (t)$ for $t\ge 0$, 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 $\alpha (t)\mathrm{/}t$ converges to a zero-mean Normal random variable as $t\to \mathrm{\infty }$) and the tail asymptotics of $P(\alpha (t)\mathrm{/}t\ge q)$. We apply our findings to spectrally positive L'evy processes reflected at the infimum and establish various new occupation time results for the corresponding model.