Occupation Time and the Lebesgue Measure of the Range for a Levy Process

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 $\lbrack 0, a \rbrack$ and $R(a)$ the Lebesgue measure of the range of the process intersected with $\lbrack 0, a \rbrack$. Whenever $\lbrack 0, \infty...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1988, Vol.103 (4), p.1241-1248
Main Author: Port, S. C.
Format: Article
Language:English
Subjects:
Brownian motion
Exact sciences and technology
Lebesgue measures
Markov processes
Mathematical theorems
Mathematics
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
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Summary: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 $\lbrack 0, a \rbrack$ and $R(a)$ the Lebesgue measure of the range of the process intersected with $\lbrack 0, a \rbrack$. Whenever $\lbrack 0, \infty)$ is a recurrent set we show $N(a)/EN(a) - R(a)/ER(a)$ converges in the mean square to 0 as $a \rightarrow \infty$. This in turn is used to derive limit laws for $R(a)/ER(a)$ from those for $N(a)/EN(a)$.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1988-0955017-X