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