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

Let $W(t)$ be a two dimensional Brownian motion with $W(0) = (1, 0)$ and let φ(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 → ∞, 4πφ(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.