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Excursions in a cone for two-dimensional Brownian motion. (English) Zbl 0582.60048
Let $$\{B(t),0\leq t<\infty \}$$ be the two-dimensional standard Brownian motion process with continuous paths on a probability space. A most significant property of the Brownian paths is known as the winding property: Let T be a finite Markov time of the two-dimensional Brownian motion process, then with probability one $$\{B(t),T\leq t<T+\epsilon \}$$ winds about B(T) and cuts itself for every $$\epsilon >0$$. In this paper we will consider the contrary. Namely, when does occur a non-winding in the two-dimensional Brownian paths? We also determine the law of a non- winding part by giving the corresponding conditioned limit theorem for the Brownian motion.

##### MSC:
 60G17 Sample path properties 60J65 Brownian motion 60F15 Strong limit theorems
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