Open Access
January 1998 Windings of Brownian motion and random walks in the plane
Zhan Shi
Ann. Probab. 26(1): 112-131 (January 1998). DOI: 10.1214/aop/1022855413


We are interested in the almost sure asymptotic behavior of the windings of planar Brownian motion. Both the usual lim sup and Chung’s lim inf versions of the law of the iterated logarithm are presented for the so-called ‘‘big’’ and ‘‘small’’ winding angles. Our method is based on some very accurate estimates of the winding clock. The corresponding problem for a spherically symmetric random walk in $\mathbb{R}^2$ is also studied. A strong approximation using the Brownian big winding process is established. Similar results are obtained.


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Zhan Shi. "Windings of Brownian motion and random walks in the plane." Ann. Probab. 26 (1) 112 - 131, January 1998.


Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0938.60073
MathSciNet: MR1617043
Digital Object Identifier: 10.1214/aop/1022855413

Primary: 60J15 , 60J65
Secondary: 60F15

Keywords: Brownian motion , Random walk , strong approximation , Winding angle

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 1 • January 1998
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