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.
"A New Proof of Spitzer's Result on the Winding of Two Dimensional Brownian Motion." Ann. Probab. 10 (1) 244 - 246, February, 1982. https://doi.org/10.1214/aop/1176993928