Let $B\_t$ be a standard one dimensional Brownian motion. Is it true that


$$\lim\_{s \to \infty} \frac{\int\_{[0, s]} \mathbf 1\_{ \{|B\_t| \geq \sqrt{2t/\pi} \} } \ dt}{s}$$


exists almost surely?



The long run average amount of time the deviation of Brownian motion spends above its expected value
The limsup is 1 (as noted by Anthony Quas) and the liminf is zero. The first of these follows immediately from Strassen's functional LIL. The liminf can be deduced from the distribution of the running maximum of Brownian motion.


[1] <https://sites.stat.washington.edu/jaw/COURSES/520s/523/HO.523.20/523-Spr2020-L4.pdf>


