The intermediate arc-sine law

• Published in: Statistics & probability letters Vol. 49; no. 2; pp. 119 - 125
• Main Authors: Nikitin, Yakov, Orsingher, Enzo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.08.2000; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Arc-sine law; Brownian bridge; Brownian motion; Brownian motion Brownian bridge Arc-sine law Second arc-sine law Sojourn time Kac empirical process; Kac empirical process; Second arc-sine law; Sojourn time; Brownian motion; Arc-sine law; Second arc-sine law; Sojourn time; Kac empirical process; Brownian bridge
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/S0167-7152(00)00038-9

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.