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

Bolthausen, E (1979). On the global asymptotic behavior of Brownian local time on the circle. Transactions of the American Mathematical Society, 253:317-328.

## 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 $t \rightarrow \infty$. 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 $S^1$.

## 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 $t \rightarrow \infty$. 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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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Brownian motion on the circle; local time; weak convergence English 1979 30 Oct 2009 14:58 05 Apr 2016 13:30 American Mathematical Society 0002-9947 https://doi.org/10.2307/1998200 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0413.60012