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

Port, S. C.

doi:10.1090/S0002-9939-1988-0955017-X

Occupation time and the Lebesgue measure of the range for a Lévy process
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by S. C. Port

Proc. Amer. Math. Soc. 103 (1988), 1241-1248

DOI: https://doi.org/10.1090/S0002-9939-1988-0955017-X

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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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