Lévy arcsine law

In probability theory, the Lévy arcsine law, found by Paul Lévy (1939), states that for a a Wiener process (which models Brownian motion) the proportion of the time that the process is positive is a random variable whose probability distribution is the arcsine distribution. That distribution has a cumulative distribution function proportional to arcsin(√x).

Suppose W is the standard Wiener process. For every T > 0, let

$m(T) = m\{\, t \in [0,T]\,:\, W(t) > 0 \,\}$

be the measure of the set of times t between 0 and T when W(t) > 0. Then for every x ∈ [0, 1],

$\Pr\left( \frac{m(T)}{T} \le x \right) = \frac{2}{\pi}\arcsin\left(\sqrt{x}\right) .$

This result is also sometimes called the "first arcsine law". The two other arcsine laws are concerned with: the time (between 0 and 1) at which W(t) attains its maximum, and the largest time t* such that W(t) remained positive after t*. There are thus three arcsine laws.

References

• Lévy, Paul (1939), "Sur certains processus stochastiques homogènes", Compositio Mathematica 7: 283–339, ISSN 0010-437X, MR 0000919 
• Rogozin, B. A. (2001), "Arcsine law", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4