# Arcsine law

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A limit theorem describing the fluctuations of a random walk on the real line, which results in an arcsine distribution or a generalized arcsine distribution. The following feature of a Brownian motion was noted in 1939 by P. Lévy. Let be the Lebesgue measure of the set or, in other words, the time spent by a Brownian particle on the positive semi-axis during the interval of time . The ratio will then have the arcsine distribution: It was subsequently noted  that a random walk with discrete time obeys the following arcsine law: Let be the successive locations in the random walk, where are independent and identically distributed, let be equal to the number of indices among for which , and let then the relationships  are all satisfied or not satisfied at the same time; here, for is the generalized arcsine distribution, where if and if .

The arcsine law in renewal theory states that for the following equalities are valid: and for where is defined by the relation , if and only if for , where is a function which is defined for and which has the property There exists a close connection between the arcsine law in renewal theory and the arcsine law governing a random walk .

How to Cite This Entry:
Arcsine law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arcsine_law&oldid=18102
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article