Windings of Brownian motion and random walks in the plane (Q1307070)

Let \(\{Z(t)= X(t)+ iY(t), t\geq 0\}\) be a planar Brownian motion starting at \(z_0\neq 0\), \(\theta(t)\) be the total angle wounded by \(Z(t)\) around origin up to time \(t\). The author considers the a.s. asymptotic behaviour of the so-called ``big windings'' \(\theta_+(t)= \int^t_0 I_{\{|Z(u)|> 1\}}d\theta(u)\) and ``small windings'' \(\theta_-(t)= \int^t_0 I_{\{|Z(u)|< 1\}}d\theta(u).\) Both \(\limsup\) and \(\liminf\) behaviour of \(\theta_+(t)\), \(\theta_-(t)\) and \(\eta(t)= a\theta_+(t)+ b\theta_-(t)\), \((a,b)\in R^2\), are investigated, the integral test for \(\eta(t)\) is proved as well as Chung-type law of the iterated logarithm. The method of proofs is based on skew-producted representation of \(Z(t)\) and accurate estimates of the Brownian winding clocks. The analogous problems for a spherically symmetric random walk \(S\) in \(R^2\) are also studied via the strong invariance principle. It is proved that windings of \(S\) behaves asymptotically as \(\theta_+(t)\). Thus \(\limsup\) and \(\liminf\) versions of LIL are obtained.