Windings of spherically symmetric random walks via Brownian embedding

Abstract

Let X1, X2, X3,... be a sequence of i.i.d. 2-valued random variables with a spherically symmetric distribution. Let (Sn; n[ges]0) be its sequence of partial sums and let ([phi](n); n[ges]0) be its winding sequence. Assuming only a mild moment condit show, via Brownian embedding, that 2[phi](n)/log n converges in distribution to a standard hyperbolic secant distribution.