Windings of Brownian motion and random walks in the plane

Abstract

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.