Perpetual Integrals for Lévy Processes

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.

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Acknowledgments

The authors thank Jean Bertoin for several discussions on the topic and suggesting the use of the Hewitt–Savage 0–1 law. The careful reading and comments of an anonymous referee are warmly acknowledged.

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Correspondence to Leif Döring.

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Leif Döring was supported by an Ambizione Grant of the Swiss Science Foundation.

Andreas E. Kyprianou acknowledges support of the Institute for Mathematical Research (FIM) at ETH Zürich and EPSRC Grant Number EP/L002442/1.

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Döring, L., Kyprianou, A.E. Perpetual Integrals for Lévy Processes. J Theor Probab 29, 1192–1198 (2016). https://doi.org/10.1007/s10959-015-0607-y

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Keywords

  • Lévy processes
  • Fluctuation theory
  • Perpetual integral

Mathematics Subject Classification

  • 60J25
  • 60J55
  • 60J75