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On the arc-sine laws for Lévy processes

Published online by Cambridge University Press:  14 July 2016

R. K. Getoor and
M. J. Sharpe
R. K. Getoor
Affiliation:
University of California, San Diego
M. J. Sharpe*
Affiliation:
University of California, San Diego
*
∗ Postal address: Department of Mathematics–0112, 9500 Gilman Drive, La Jolla, CA 92093–0112, USA.
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Abstract

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 tP 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs>0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs>0} ds under P 0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

Keywords

SPITZER'S THEOREMRANDOM WALKS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994, pp. 76 - 89
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported in part by NSF Grant DMS 91–01675.

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