The intermediate arc-sine law

It is well-known that the sojourn time of Brownian motion B(t), t>0, namely [Gamma](B)=meas(s[less-than-or-equals, slant]1: B(s)>0), obeys the arc-sine law, while, subject to the condition B(1)=0, is uniformly distributed. We present here the distribution of [Gamma](B) under the condition B(u)=0, for u[greater-or-equal, slanted]1. This is called the intermediate arc-sine law and it is shown that it converges to the classical one as u-->[infinity] and becomes the uniform law as u=1. We also show that the first instant where the maximum of Brownian motion is attained follows the intermediate arc-sine law when the condition B(u)=0, u[greater-or-equal, slanted]1, is assumed. It is pointed out that such "intermediate" arc-sine laws are connected with generalized Kac empirical processes.Brownian motion Brownian bridge Arc-sine law Second arc-sine law Sojourn time Kac empirical process