Abstract
If $\cos(\theta/2) < 1/\sqrt n$ then a.s. there are times $0 \leq s_1 < s_2$ such that the $n$-dimensional Brownian motion $Z(t)$ stays for all $t \in (s_1, s_2)$ in a cone with vertex $Z(s_1)$ and angle $\theta$. If $\cos(\theta/2) > 1/\sqrt n$ then the same event has probability 0.
Citation
Krzysztof Burdzy. "Brownian Paths and Cones." Ann. Probab. 13 (3) 1006 - 1010, August, 1985. https://doi.org/10.1214/aop/1176992922
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