The Annals of Probability

A New Proof of Spitzer's Result on the Winding of Two Dimensional Brownian Motion

Richard Durrett

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Abstract

Let $W(t)$ be a two dimensional Brownian motion with $W(0) = (1, 0)$ and let $\varphi(t)$ be the net number of times the path has wound around (0, 0), counting clockwise loops as $-1$, counterclockwise as $+1$. Spitzer has shown that as $t \rightarrow \infty, 4\pi\varphi(t)/\log t$ converges to a Cauchy distribution with parameter 1. In this paper we will use Levy's result on the conformal invariance of Brownian motion to give a simple proof of Spitzer's theorem.

Article information

Source
Ann. Probab., Volume 10, Number 1 (1982), 244-246.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993928

Digital Object Identifier
doi:10.1214/aop/1176993928

Mathematical Reviews number (MathSciNet)
MR637391

Zentralblatt MATH identifier
0479.60081

JSTOR
links.jstor.org

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60F05: Central limit and other weak theorems

Keywords
Brownian motion winding Cauchy distribution Levy's theorem

Citation

Durrett, Richard. A New Proof of Spitzer's Result on the Winding of Two Dimensional Brownian Motion. Ann. Probab. 10 (1982), no. 1, 244--246. doi:10.1214/aop/1176993928. https://projecteuclid.org/euclid.aop/1176993928


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