Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] H. Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447-456. MR 0178494 (31:2751)
  • [2] F. B. Knight, Random walks and a sojourn density process of Brownian motion, Trans. Amer. Math. Soc. 109 (1963), 56-86. MR 0154337 (27:4286)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60J55, 60F15, 60J65

Retrieve articles in all journals with MSC: 60J55, 60F15, 60J65


Additional Information

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

American Mathematical Society