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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A problem of Földes
and Puri on the Wiener process


Author: Z. Shi
Journal: Trans. Amer. Math. Soc. 348 (1996), 219-228
MSC (1991): Primary 60J65; Secondary 60G17
MathSciNet review: 1321589
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Abstract: Let $W$ be a real-valued Wiener process starting from 0, and $\tau (t)$ be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of $X(t)=\int _0^{\tau (t)} {\text{\bf 1}\hskip -1.25pt\mathrm{l}}_{\{ | W(u)| \le \alpha t\} }du$ as $t$ tends to infinity, i.e. they ask: how long does $W$ stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for $X$.


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Additional Information

Z. Shi
Email: shi@ccr.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01485-7
PII: S 0002-9947(96)01485-7
Keywords: Wiener process (Brownian motion), law of the iterated logarithm
Received by editor(s): December 7, 1994
Article copyright: © Copyright 1996 American Mathematical Society