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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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