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

S. C. Port
Proceedings of the American Mathematical Society
Vol. 103, No. 4 (Aug., 1988), pp. 1241-1248
DOI: 10.2307/2047120
Stable URL: http://www.jstor.org/stable/2047120
Page Count: 8
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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 $\lbrack 0, a \rbrack$ 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 $N(a)/EN(a)$.

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