# Lévy arcsine law

In probability theory, the **Lévy arcsine law**, found by Template:Harvs, states that the probability distribution of the proportion of the time that a Wiener process (which models Brownian motion) 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.

## References

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