Arcsine laws (Wiener process)


In probability theory, the arcsine laws are a collection of results for one-dimensional random walks and Brownian motion. The best known of these is attributed to.
All three laws relate path properties of the Wiener process to the arcsine distribution. A random variable X on is arcsine-distributed if

Statement of the laws

Throughout we suppose that.

First (Lévy's) arcsine law

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 at which the Wiener process is positive. Then is arcsine distributed

Second arcsine law

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.

Third arcsine law

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.

Equivalence of the second and third laws

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|.
Since the zeros of B and |B| coincide, the last zero of X has the same distribution as L, the last zero of the Wiener process. The last zero of X occurs exactly when W achieves its maximum. It follows that the second and third laws are equivalent.