## The Annals of Probability

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

Richard Durrett

#### 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

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

Subjects
Secondary: 60F05: Central limit and other weak theorems

#### 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