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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 $0<x<1$, $$\mathbb P(A(t)/t\leqslant x) = \frac2\pi \arcsin \sqrt x. $$

How can this be proven?

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    $\begingroup$ 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... $\endgroup$ – saz May 14 at 20:09
  • $\begingroup$ @saz Thanks for the reference. $\endgroup$ – Math1000 May 14 at 20:41

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