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:
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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,
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where are independent and identically distributed, let
be equal to the number of indices
among
for which
, and let
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then the relationships
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are all satisfied or not satisfied at the same time; here, for
is the generalized arcsine distribution,
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where if
and
if
.
The arcsine law in renewal theory states that for the following equalities are valid:
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and for
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where is defined by the relation
,
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if and only if
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for , where
is a function which is defined for
and which has the property
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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.
Arcsine law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arcsine_law&oldid=25962