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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the iterated logarithm law for local time


Author: Edwin Perkins
Journal: Proc. Amer. Math. Soc. 81 (1981), 470-472
MSC: Primary 60J55; Secondary 60F15, 60J65
DOI: https://doi.org/10.1090/S0002-9939-1981-0597665-9
MathSciNet review: 597665
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Abstract: If $ s(t,x)$ is the local time of a Brownian motion, we show that $ \theta (\alpha ) = \lim {\sup _{t \to \infty }}{\inf _{\left\vert x \right\ver... ...t \alpha {t^{1/2}}{{(2\log \log t)}^{ - 1/2}}}}s(t,x){(2t\log \log t)^{ - 1/2}}$ satisfies

$\displaystyle {((1 - {\alpha ^{1/2}}) \vee 0)^2} \leqslant \theta (\alpha ) \leqslant {(2\alpha )^{ - 1}} \wedge 1.$

In particular, it follows from a result of Kesten that

$\displaystyle \mathop {\lim \sup }\limits_{t \to \infty } s(t,x){(2t\log \log t)^{ - 1/2}} = 1$

for all $ x$ a.s.

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DOI: https://doi.org/10.1090/S0002-9939-1981-0597665-9
Article copyright: © Copyright 1981 American Mathematical Society