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.
Citation
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. https://doi.org/10.1214/aop/1176992529
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