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Proceedings of the American Mathematical Society

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Occupation time and the Lebesgue measure of the range for a Lévy process


Author: S. C. Port
Journal: Proc. Amer. Math. Soc. 103 (1988), 1241-1248
MSC: Primary 60J30
MathSciNet review: 955017
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Abstract: We consider a Levy process on the line that is transient and with nonpolar one point sets. For $ a > 0$ let $ N(a)$ be the total occupation time of $ [0,a]$ and $ R(a)$ the Lebesgue measure of the range of the process intersected with $ [0,a]$. Whenever $ [0,\infty ]$ is a recurrent set we show $ N(a)/EN(a) - R(a)/ER(a)$ converges in the mean square to 0 as $ a \to \infty $. This in turn is used to derive limit laws for $ R(a)/ER(a)$ from those for $ N(a)/EN(a)$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0955017-X
Keywords: Levy process, I.D. process, occupation time, range
Article copyright: © Copyright 1988 American Mathematical Society