probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the arcsine laws are a collection of results for one-dimensional

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

s and Brownian motion (the

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...

). The best known of these is attributed to . All three laws relate path properties of the Wiener process to the

arcsine distribution In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: :F(x) = \frac\arcsin\left(\sqrt x\right)=\frac+\frac for 0 ≤ ''x'' ...

. A random variable ''X'' on ,1is arcsine-distributed if :

\Pr \left X \leq x \right = \frac \arcsin\left(\sqrt\right), \qquad \forall x \in,1

Statement of the laws

Throughout we suppose that (''W''_''t'')_{0 ≤ ''t'' ≤ 1} ∈ R is the one-dimensional Wiener process on ,1 Scale invariance ensures that the results can be generalised to Wiener processes run for ''t'' ∈ \\ be the Lebesgue measure, measure of the set of times in ,1at which the Wiener process is positive. Then

T_+

is arcsine distributed.

Second arcsine law

The second arcsine law describes the distribution of the last time the Wiener process changes sign. Let :

L=\sup\left\

be the 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,Morters, Peter and Peres, Yuval, ''Brownian Motion'', Chapter 2. 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 :

W_M = \sup \.

Then ''M'' is arcsine distributed.

Equivalence of the second and third laws

Defining the running maximum process ''M''_''t'' of the Wiener process :

M_t = \sup \,

then the law of ''X''_''t'' = ''M''_''t'' − ''W''_''t'' has the same law as a reflected Wiener process , ''B''_''t'', (where ''B''_''t'' is a Wiener process independent of ''W''_''t''). 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.

Notes

References

* * * {{eom, id=A/a013170, first=B. A., last= Rogozin, title=Arcsine law Wiener process Statistical mechanics