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April, 1986 How Small are the Increments of the Local Time of a Wiener Process?
E. Csaki, A. Foldes
Ann. Probab. 14(2): 533-546 (April, 1986). DOI: 10.1214/aop/1176992529


Let $W(t)$ be a standard Wiener process with local time $L(x, t)$. Put $L(t) = L(0, t)$ and $L^\ast(t) = \sup_{-\infty < x < \infty} L(x, t)$. We study the almost sure behaviour of small increments of $L(t)$ and also, the joint behaviour of $L(t)$ and the last excursion, $U(t)$. The increment problem of $L(x, t)$ are also studied uniformly in $x$. This implies a $\lim \inf$-type law of the iterated logarithm for $L^\ast(t)$ due to Kesten (1965), in which case the exact constant, not known before, is also determined.


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E. Csaki. A. Foldes. "How Small are the Increments of the Local Time of a Wiener Process?." Ann. Probab. 14 (2) 533 - 546, April, 1986.


Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0598.60083
MathSciNet: MR832022
Digital Object Identifier: 10.1214/aop/1176992529

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

Keywords: integral tests , Local time , small increments of Brownian local time , Wiener process (Brownian motion)

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 2 • April, 1986
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