The intermediate arc-sine law

Yakov Nikitin and Enzo Orsingher

Statistics & Probability Letters, 2000, vol. 49, issue 2, 119-125

Abstract: 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.

Keywords: Brownian; motion; Brownian; bridge; Arc-sine; law; Second; arc-sine; law; Sojourn; time; Kac; empirical; process (search for similar items in EconPapers)
Date: 2000
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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