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

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 {S^1} . It is shown that after appropriate norming the distribution of this random element in C ( S 1 ) C({S^1}) converges weakly as t → ∞ t\, \to \,\infty . The limit is identified as 2 ( B ( x ) − ∫ B ( y ) d y ) 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 {S^1} .