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

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

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

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

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

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