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Difference between revisions of "Arcsine law"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Spitzer,   "Principles of random walk" , Springer (1976)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Spitzer, "Principles of random walk", Springer (1976)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317035.png" /> in the article above is called a slowly varying function, cf. [[#References|[1]]], p. 269.
 
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317035.png" /> in the article above is called a slowly varying function, cf. [[#References|[1]]], p. 269.

Revision as of 11:22, 4 May 2012

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.

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