Elementary derivation of Spitzer's asymptotic law for Brownian windings and some of its physical applications

A simple derivation of Spitzer’z asymptotic law for Brownian windings [Trans.Am.Math.Soc.87,187 (1958)]is presented along with its generalizations

.These include the cases of planar Brownian walks interacting with a single puncture and Brownian walks on a single truncated cone with variable conical angle interacting with the truncated conical tip.Such situations are typical in the theories of quantum Hall effect and 2+1 quantum gravity, respectively .They also have some applications in polymer physics