Paul Lévy’s Arcsine Laws

  • Ju-Yi Yen
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 2088)


Lévy proved that both the last zero of Brownian motion before time 1, and the time spent positive unto time 1 by Brownian motion are arc-sine distributed. We first recall some elementary representations of an arc-sine distributed variable. Lévy’s first result derives easily from there. As to Lévy’s second result, it follows from Tanaka’s formula used to write the positive and negative parts of Brownian motion as two independent reflecting Brownian motions, time-changed. We also study random Brownian scaling, which yields in particular another result of Lévy: the time spent positive by the Brownian bridge is uniformly distributed on (0, 1).


Brownian Motion Random Time Normal Random Variable Brownian Bridge Standard Normal Random Variable 
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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Ju-Yi Yen
    • 1
  • Marc Yor
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris VIParis CX 05France

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