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The intermediate arc-sine law

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  • Nikitin, Yakov
  • Orsingher, Enzo

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.

Suggested Citation

  • Nikitin, Yakov & Orsingher, Enzo, 2000. "The intermediate arc-sine law," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 119-125, August.
    • Handle: RePEc:eee:stapro:v:49:y:2000:i:2:p:119-125
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    Cited by:

    1. Iafrate, F. & Orsingher, E., 2021. "On the sojourn time of a generalized Brownian meander," Statistics & Probability Letters, Elsevier, vol. 168(C).
    2. Beghin, L. & Nikitin, Y. & Orsingher, E., 2003. "How the sojourn time distributions of Brownian motion are affected by different forms of conditioning," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 291-302, December.

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