The Annals of Probability

How Small are the Increments of the Local Time of a Wiener Process?

E. Csaki and A. Foldes

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Abstract

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.

Article information

Source
Ann. Probab., Volume 14, Number 2 (1986), 533-546.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992529

Digital Object Identifier
doi:10.1214/aop/1176992529

Mathematical Reviews number (MathSciNet)
MR832022

Zentralblatt MATH identifier
0598.60083

JSTOR
links.jstor.org

Subjects
Primary: 60J55: Local time and additive functionals
Secondary: 60J65: Brownian motion [See also 58J65] 60G17: Sample path properties 60G57: Random measures

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

Citation

Csaki, E.; Foldes, A. How Small are the Increments of the Local Time of a Wiener Process?. Ann. Probab. 14 (1986), no. 2, 533--546. doi:10.1214/aop/1176992529. https://projecteuclid.org/euclid.aop/1176992529


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