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August, 1983 How Big are the Increments of the Local Time of a Wiener Process?
E. Csaki, M. Csorgo, A. Foldes, P. Revesz
Ann. Probab. 11(3): 593-608 (August, 1983). DOI: 10.1214/aop/1176993504


Let $W(t)$ be a standard Wiener process with local time (occupation density) $L(x, t)$. Kesten showed that $L(0, t)$ and $\sup_x L(x, t)$ have the same LIL law as $W(t)$. Exploiting a famous theorem of P. Levy, namely that $\{\sup_{0 \leq s \leq t} W(s), t \geq 0\} {\underline{\underline{\mathscr{D}}}} \{L(0, t), t \geq 0\}$, we study the almost sure behaviour of big increments of $L(0, t)$ in $t$. The very same increment problems in $t$ of $L(x, t)$ are also studied uniformly in $x$. The results in the latter case are slightly different from those concerning $L(0, t)$, and they coincide only for Kesten's above mentioned LIL.


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E. Csaki. M. Csorgo. A. Foldes. P. Revesz. "How Big are the Increments of the Local Time of a Wiener Process?." Ann. Probab. 11 (3) 593 - 608, August, 1983.


Published: August, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0545.60074
MathSciNet: MR704546
Digital Object Identifier: 10.1214/aop/1176993504

Primary: 60J55
Secondary: 60G17 , 60G57 , 60J65

Keywords: big increments and continuity of Brownian local time , Local time , Wiener process (Brownian motion)

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 3 • August, 1983
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