## Étude asymptotique du temps passé par le mouvement brownien dans un cône. (Asymptotics for the time spent by Brownian motion in a cone).(French)Zbl 0727.60092

Summary: Let $${\mathcal T}(t)$$ be the total time spent, before time t, by a d-dimensional Brownian motion started at 0 in a closed cone of $${\mathbb{R}}^ d$$ with vertex 0, based on an arbitrary subset of the sphere. We discuss the asymptotic behavior, at $$t=0$$ and $$t=+\infty$$, of the process $$(| \log t|^ q{\mathcal T}(t)/t,\quad t>0),$$ according to the value of the parameter $$q>0$$. The proof uses certain estimates for the distribution function of the hitting time of a cone for a Brownian motion started at $$\alpha \in {\mathbb{R}}^ d-\{0\}$$.

### MSC:

 60J65 Brownian motion 60G17 Sample path properties 60F05 Central limit and other weak theorems
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