## Windings of Brownian motion and random walks in the plane.(English)Zbl 0938.60073

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.

### MSC:

 60J65 Brownian motion 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems
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### References:

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