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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the global asymptotic behavior of Brownian local time on the circle

Author: E. Bolthausen
Journal: Trans. Amer. Math. Soc. 253 (1979), 317-328
MSC: Primary 60F05; Secondary 60J55
MathSciNet review: 536950
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Abstract: The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point t this is a (random) continuous function on $ {S^1}$. It is shown that after appropriate norming the distribution of this random element in $ C({S^1})$ converges weakly as $ t\, \to \,\infty $. The limit is identified as $ 2(B(x)\, - \,\int {B(y)\,dy)} $ where B is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant t on $ {S^1}$.

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Keywords: Brownian motion on the circle, local time, weak convergence
Article copyright: © Copyright 1979 American Mathematical Society