# How to prove arcsine law for amount of time Brownian motion is positive in $[0,t]$?

Ross's Introduction to Stochastic Processes states, but does not prove this result:

For Brownian motion, let $$A(t)$$ denote the amount of time in $$[0,t]$$ the process is positive. Then, for $$0, $$\mathbb P(A(t)/t\leqslant x) = \frac2\pi \arcsin \sqrt x.$$

How can this be proven?

• Take e.g. a look at Application 8.7 in Brownian motion - An introduction to stochastic processes by Schilling & Partzsch. Pretty sure that there are other books which contain the proof as well... – saz May 14 at 20:09
• @saz Thanks for the reference. – Math1000 May 14 at 20:41