Abstract
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).
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Yen, JY., Yor, M. (2013). Paul Lévy’s Arcsine Laws. In: Local Times and Excursion Theory for Brownian Motion. Lecture Notes in Mathematics, vol 2088. Springer, Cham. https://doi.org/10.1007/978-3-319-01270-4_4
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DOI: https://doi.org/10.1007/978-3-319-01270-4_4
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