Arcsine law

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 [2] 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 [3].

References

[1]	W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)
[2]	F. Spitzer, "Principles of random walk", Springer (1976)
[3]	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

Comments

The function in the article above is called a slowly varying function, cf. [1], p. 269.