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:
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 .
|||W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)|
|||F. Spitzer, "Principles of random walk", Springer (1976)|
|||B.A. Rogozin, "The distribution of the first ladder moment and height and fluctuation of a random walk" Theory Probab. Appl., 16 : 4 (1971) pp. 575–595 Teor. Veroyatnost. i Primenen., 16 : 4 (1971) pp. 593–613|
The function in the article above is called a slowly varying function, cf. , p. 269.
Arcsine law. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arcsine_law&oldid=25962