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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 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 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
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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