Lévy arcsine law
In probability theory, the arcsine laws are a collection of results for one-dimensional random walks and Brownian motion (the Wiener process). The best known of these is attributed to Paul Lévy (1939).
All three laws relate path properties of the Wiener process to the arcsine distribution. A random variable X on [0,1] is arcsine-distributed if
Throughout we suppose that (Wt)0 ≤ t ≤ 1 ∈ R is the one-dimensional Wiener process on [0,1]. Scale invariance ensures that the results can be generalised to Wiener processes run for t ∈[0,∞).
The first arcsine law states that the proportion of time that the one-dimensional Wiener process is positive follows an arcsine distribution. Let
be the measure of the set of times in [0,1] at which the Wiener process is positive. Then is arcsine distributed
The second arcsine law describes the distribution of the last time the Wiener process changes sign. Let
be the last time of the last zero. Then L is arcsine distributed.
The third arcsine law states that the time at which a Wiener process achieves its maximum is arcsine distributed.
The statement of the law relies on the fact that the Wiener process has an almost surely unique maxima, and so we can define the random variable M which is the time at which the maxima is achieved. i.e. the unique M such that
Then M is arcsine distributed.
Defining the running maximum process Mt of the Wiener process
then the law of Xt = Mt − Wt has the same law as a reflected Wiener process |Bt| (where Bt is a Wiener process independent of Wt).
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