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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|arcsine distribution]]. The following feature of a Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131701.png" /> was noted in 1939 by P. Lévy. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131702.png" /> be the Lebesgue measure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131703.png" /> or, in other words, the time spent by a Brownian particle on the positive semi-axis during the interval of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131704.png" />. The ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131705.png" /> will then have the arcsine distribution:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131706.png" /></td> </tr></table>
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{{TEX|done}}
  
It was subsequently noted [[#References|[2]]] that a random walk with discrete time obeys the following arcsine law: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131707.png" /> be the successive locations in the random walk,
+
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|arcsine distribution]]. The following feature of a Brownian motion  $  \{ {\xi _ {t} } : {t \geq  0, \xi _ {0} =0 } \} $
 +
was noted in 1939 by P. Lévy. Let  $  \tau _ {t} $
 +
be the Lebesgue measure of the set  $  \{ {u } : {\xi _ {u} > 0, 0 \leq  u \leq  t } \} $
 +
or, in other words, the time spent by a Brownian particle on the positive semi-axis during the interval of time  $  [0, t] $.
 +
The ratio  $  \tau _ {t} / t $
 +
will then have the arcsine distribution:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131708.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{
 +
\frac{\tau _ {t} }{t}
 +
< x \right \}  = \
 +
F _ {1/2} (x)  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a0131709.png" /> are independent and identically distributed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317010.png" /> be equal to the number of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317011.png" /> among <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317013.png" />, and let
+
\frac{2} \pi
 +
  \mathop{\rm arcsin} \
 +
\sqrt {x } ,\  0 \leq  x \leq  1,\  t > 0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317014.png" /></td> </tr></table>
+
It was subsequently noted [[#References|[2]]] that a random walk with discrete time obeys the following arcsine law: Let  $  S _ {0} = 0 \dots S _ {n} \dots $
 +
be the successive locations in the random walk,
 +
 
 +
$$
 +
S _ {n}  = \sum _ {u = 1 } ^ { n }
 +
\xi _ {u} ,
 +
$$
 +
 
 +
where  $  \xi _ {1} , \xi _ {2} \dots $
 +
are independent and identically distributed, let  $  T _ {n} $
 +
be equal to the number of indices  $  k $
 +
among  $  0 \dots n $
 +
for which  $  S _ {k} > 0 $,
 +
and let
 +
 
 +
$$
 +
K _ {n}  =   \mathop{\rm min} \left \{ {k } : {S _ {k} =
 +
\max _ {0 \leq  m \leq  n }  S _ {m} } \right \}
 +
,
 +
$$
  
 
then the relationships
 
then the relationships
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317015.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{
 +
\frac{T _ {n} }{n}
 +
< x \right \}  = \
 +
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{
 +
\frac{K _ {n} }{n}
 +
< x \right \}
 +
= F _ {a} (x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317016.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } 
 +
\frac{ {\mathsf P} \{ S _ {1} < 0 \} + \dots + {\mathsf P} \{ S _ {n} < 0 \} }{n}
 +
  = \alpha
 +
$$
  
are all satisfied or not satisfied at the same time; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317018.png" /> is the generalized arcsine distribution,
+
are all satisfied or not satisfied at the same time; here, $  {F _  \alpha  } (x) $
 +
for  $  0 < \alpha < 1 $
 +
is the generalized arcsine distribution,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317019.png" /></td> </tr></table>
+
$$
 +
F _ {1} (x)  =   {\mathsf E}  (x) \  \textrm{ and } \  F _ {0} (x)  = \
 +
{\mathsf E}  ( x - 1 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317023.png" />.
+
where $  {\mathsf E}  (x) = 0 $
 +
if $  x \leq  0 $
 +
and $  {\mathsf E}  (x) = 1 $
 +
if  $  x > 0 $.
  
The arcsine law in renewal theory states that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317024.png" /> the following equalities are valid:
+
The arcsine law in renewal theory states that for $  0 < \alpha < 1 $
 +
the following equalities are valid:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317025.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ \xi _ {1} \geq  0 \}  = 1
 +
$$
  
 
and for
 
and for
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317026.png" /></td> </tr></table>
+
$$
 +
y _ {t}  = t - S _ {\eta _ {t}  } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317027.png" /> is defined by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317028.png" />,
+
where $  \eta _ {t} $
 +
is defined by the relation $  {S _ {\eta _ {t}  } } < t \leq  S _ {\eta _ {t + 1 }  } $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317029.png" /></td> </tr></table>
+
$$
 +
\lim\limits  {\mathsf P} \left \{
 +
\frac{y _ {t} }{t}
 +
< x \right \}  = \
 +
F _  \alpha  (x)
 +
$$
  
 
if and only if
 
if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317030.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ \xi _ {1} > x \}  = x ^ {- \alpha } L (x)
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317032.png" /> is a function which is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317033.png" /> and which has the property
+
for $  x > 0 $,  
 +
where $  L(x) $
 +
is a function which is defined for $  x > 0 $
 +
and which has the property
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013170/a01317034.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow \infty } 
 +
\frac{L (xy) }{L (x) }
 +
  = 1 \ \
 +
\textrm{ for } \  0 < y < \infty .
 +
$$
  
 
There exists a close connection between the arcsine law in renewal theory and the arcsine law governing a random walk [[#References|[3]]].
 
There exists a close connection between the arcsine law in renewal theory and the arcsine law governing a random walk [[#References|[3]]].
Line 49: Line 129:
  
 
====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 $  L $
 +
in the article above is called a slowly varying function, cf. [[#References|[1]]], p. 269.

Latest revision as of 18:48, 5 April 2020


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 $ \{ {\xi _ {t} } : {t \geq 0, \xi _ {0} =0 } \} $ was noted in 1939 by P. Lévy. Let $ \tau _ {t} $ be the Lebesgue measure of the set $ \{ {u } : {\xi _ {u} > 0, 0 \leq u \leq t } \} $ or, in other words, the time spent by a Brownian particle on the positive semi-axis during the interval of time $ [0, t] $. The ratio $ \tau _ {t} / t $ will then have the arcsine distribution:

$$ {\mathsf P} \left \{ \frac{\tau _ {t} }{t} < x \right \} = \ F _ {1/2} (x) = \ \frac{2} \pi \mathop{\rm arcsin} \ \sqrt {x } ,\ 0 \leq x \leq 1,\ t > 0. $$

It was subsequently noted [2] that a random walk with discrete time obeys the following arcsine law: Let $ S _ {0} = 0 \dots S _ {n} \dots $ be the successive locations in the random walk,

$$ S _ {n} = \sum _ {u = 1 } ^ { n } \xi _ {u} , $$

where $ \xi _ {1} , \xi _ {2} \dots $ are independent and identically distributed, let $ T _ {n} $ be equal to the number of indices $ k $ among $ 0 \dots n $ for which $ S _ {k} > 0 $, and let

$$ K _ {n} = \mathop{\rm min} \left \{ {k } : {S _ {k} = \max _ {0 \leq m \leq n } S _ {m} } \right \} , $$

then the relationships

$$ \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \frac{T _ {n} }{n} < x \right \} = \ \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \frac{K _ {n} }{n} < x \right \} = F _ {a} (x) , $$

$$ \lim\limits _ {n \rightarrow \infty } \frac{ {\mathsf P} \{ S _ {1} < 0 \} + \dots + {\mathsf P} \{ S _ {n} < 0 \} }{n} = \alpha $$

are all satisfied or not satisfied at the same time; here, $ {F _ \alpha } (x) $ for $ 0 < \alpha < 1 $ is the generalized arcsine distribution,

$$ F _ {1} (x) = {\mathsf E} (x) \ \textrm{ and } \ F _ {0} (x) = \ {\mathsf E} ( x - 1 ) , $$

where $ {\mathsf E} (x) = 0 $ if $ x \leq 0 $ and $ {\mathsf E} (x) = 1 $ if $ x > 0 $.

The arcsine law in renewal theory states that for $ 0 < \alpha < 1 $ the following equalities are valid:

$$ {\mathsf P} \{ \xi _ {1} \geq 0 \} = 1 $$

and for

$$ y _ {t} = t - S _ {\eta _ {t} } , $$

where $ \eta _ {t} $ is defined by the relation $ {S _ {\eta _ {t} } } < t \leq S _ {\eta _ {t + 1 } } $,

$$ \lim\limits {\mathsf P} \left \{ \frac{y _ {t} }{t} < x \right \} = \ F _ \alpha (x) $$

if and only if

$$ {\mathsf P} \{ \xi _ {1} > x \} = x ^ {- \alpha } L (x) $$

for $ x > 0 $, where $ L(x) $ is a function which is defined for $ x > 0 $ and which has the property

$$ \lim\limits _ {x \rightarrow \infty } \frac{L (xy) }{L (x) } = 1 \ \ \textrm{ for } \ 0 < y < \infty . $$

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 $ L $ 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=25962
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article