The intermediate arc-sine law - Nikitin, Yakov; Orsingher, Enzo (Statistics & probability letters, 2000)
It is well-known that the sojourn time of Brownian motion B(t), t>0, namely Γ(B)= meas(s⩽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 Γ( B) under the condition B( u)=0, for u⩾1. This is called the intermediate arc-sine law and it is shown that it converges to the classical one as u→∞ 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⩾1, is assumed. It is pointed out that such “intermediate” arc-sine laws are connected with generalized Kac empirical processes.