Prefer a chat interface with context about you and your work?
Brownian Paths and Cones
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.