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

Bolthausen, E.

doi:10.1090/S0002-9947-1979-0536950-9

On the global asymptotic behavior of Brownian local time on the circle
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Trans. Amer. Math. Soc. 253 (1979), 317-328 Request permission

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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