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

TL;DR

This paper characterizes the occupation time of processes, including Lévy processes, providing exact formulas, limit theorems, and tail asymptotics, with applications to reflected spectrally positive Lévy processes.

Contribution

It offers new exact characterizations, limit theorems, and tail asymptotics for occupation times of Lévy processes, including applications to reflected processes.

Findings

Exact transform characterizations of occupation times

Central limit theorem for occupation time ratios

Tail asymptotics of occupation time probabilities

Abstract

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 $t\geq0$, 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 $\alpha(t)/t$ converges to a zero-mean Normal random variable as $t\rightarrow\infty$) and the tail asymptotics of $P(\alpha(t)/t\geq q)$. 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.