Perpetual integrals for Lévy processes

Leif Doering, Andreas E. Kyprianou

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law $$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$ with the exact characterization $$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$ The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.
Original languageEnglish
Pages (from-to)1192-1198
Number of pages7
JournalJournal of Theoretical Probability
Volume29
Issue number3
Early online date7 Mar 2015
DOIs
Publication statusPublished - Sep 2016

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