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''Brownian excursion process''
 
''Brownian excursion process''
  
The limiting process of a [[Bernoulli excursion|Bernoulli excursion]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109401.png" /> is a Bernoulli excursion, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109402.png" />, then the finite-dimensional distributions of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109403.png" /> converge to the corresponding finite-dimensional distributions of a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109404.png" /> which is called a Brownian excursion (process). The Brownian excursion process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109405.png" /> is a [[Markov process|Markov process]] for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109407.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109408.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b1109409.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094010.png" /> has a density function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094011.png" />. Obviously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094015.png" />, then
+
The limiting process of a [[Bernoulli excursion|Bernoulli excursion]]. If $  \{ \eta _ {0} , \dots, \eta _ {2n }  \} $
 +
is a Bernoulli excursion, and if $  n \rightarrow \infty $,  
 +
then the finite-dimensional distributions of the process $  \{ { {{\eta _ {[ 2nt ] }  } / {\sqrt {2n } } } } : {0 \leq  t \leq  1 } \} $
 +
converge to the corresponding finite-dimensional distributions of a process $  \{ {\eta ( t ) } : {0 \leq  t \leq  1 } \} $
 +
which is called a Brownian excursion (process). The Brownian excursion process $  \{ {\eta ( t ) } : {0 \leq  t \leq  1 } \} $
 +
is a [[Markov process|Markov process]] for which $  {\mathsf P} \{ \eta ( 0 ) = 0 \} = {\mathsf P} \{ \eta ( 1 ) = 0 \} = 1 $
 +
and $  {\mathsf P} \{ \eta ( t ) \geq  0 \} = 1 $
 +
for 0 \leq  t \leq  1 $.  
 +
If $  0 < t < 1 $,  
 +
then $  \eta ( t ) $
 +
has a density function $  f ( t,x ) $.  
 +
Obviously, $  f ( t,x ) = 0 $
 +
for $  x \leq  0 $.  
 +
If $  0 < t < 1 $
 +
and  $  x > 0 $,  
 +
then
  
<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/b/b110/b110940/b11094016.png" /></td> </tr></table>
+
$$
 +
f ( t,x ) = {
 +
\frac{2x  ^ {2} }{\sqrt {2 \pi t  ^ {3} ( 1 - t )  ^ {3} } }
 +
} e ^ {- { {x  ^ {2} } / {( 2t ( 1 - t ) ) } } } .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094017.png" />, then the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094019.png" /> have a joint density function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094020.png" />. One finds that<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094026.png" />, then
+
If $  0 < t < u < 1 $,  
 +
then the random variables $  \eta ( t ) $
 +
and $  \eta ( u ) $
 +
have a joint density function $  f ( t,x;u,y ) $.  
 +
One finds that $  f ( t,x;u,y ) = 0 $
 +
if $  x \leq  0 $
 +
or $  y \leq  0 $.  
 +
If $  0 < t < u < 1 $
 +
and  $  x > 0 $,
 +
$  y > 0 $,  
 +
then
  
<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/b/b110/b110940/b11094027.png" /></td> </tr></table>
+
$$
 +
f ( t,x;u,y ) =
 +
$$
  
<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/b/b110/b110940/b11094028.png" /></td> </tr></table>
+
$$
 +
=  
 +
{
 +
\frac{\sqrt {8 \pi } xy }{\sqrt {t  ^ {3} ( u - t ) ( 1 - u )  ^ {3} } }
 +
} \phi \left ( {
 +
\frac{x}{\sqrt t }
 +
} \right ) \phi \left ( {
 +
\frac{y}{\sqrt {1 - u } }
 +
} \right )  \times
 +
$$
  
<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/b/b110/b110940/b11094029.png" /></td> </tr></table>
+
$$
 +
\times
 +
\left [ \phi \left ( {
 +
\frac{x - y }{\sqrt {u - t } }
 +
} \right ) - \phi \left ( {
 +
\frac{x + y }{\sqrt {u - t } }
 +
} \right ) \right ] ,
 +
$$
  
 
where
 
where
  
<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/b/b110/b110940/b11094030.png" /></td> </tr></table>
+
$$
 +
\phi ( x ) = {
 +
\frac{1}{\sqrt {2 \pi } }
 +
} e ^ { {{x  ^ {2} } / 2 } }
 +
$$
  
is the normal density function (cf. [[Normal distribution|Normal distribution]]; [[Density of a probability distribution|Density of a probability distribution]]). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094031.png" /> is a Markov process, the density functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094033.png" /> completely determine its finite-dimensional distributions. For various properties of the Brownian excursion process, see [[#References|[a3]]], [[#References|[a1]]], [[#References|[a4]]].
+
is the normal density function (cf. [[Normal distribution|Normal distribution]]; [[Density of a probability distribution|Density of a probability distribution]]). Since $  \{ {\eta ( t ) } : {0 \leq  t \leq  1 } \} $
 +
is a Markov process, the density functions $  f ( t,x ) $
 +
and $  f ( t,x;u,y ) $
 +
completely determine its finite-dimensional distributions. For various properties of the Brownian excursion process, see [[#References|[a3]]], [[#References|[a1]]], [[#References|[a4]]].
  
The distributions of various functionals of the Brownian excursion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094034.png" /> frequently appear in [[Probability theory|probability theory]]. In particular, many limit distributions of the Bernoulli excursion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094035.png" /> can be expressed simply as the distributions of certain functionals of the Brownian excursion. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094036.png" />, then
+
The distributions of various functionals of the Brownian excursion $  \{ {\eta ( t ) } : {0 \leq  t \leq  1 } \} $
 +
frequently appear in [[Probability theory|probability theory]]. In particular, many limit distributions of the Bernoulli excursion $  \{ \eta _ {0} \dots \eta _ {2n }  \} $
 +
can be expressed simply as the distributions of certain functionals of the Brownian excursion. For example, if $  \delta _ {n} = \max  ( \eta _ {0} , \dots, \eta _ {2n }  ) $,  
 +
then
  
<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/b/b110/b110940/b11094037.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {n \rightarrow \infty } {\mathsf P} \left \{ {
 +
\frac{\delta _ {n} }{\sqrt {2n } }
 +
} \leq  x \right \} = F ( x ) ,
 +
$$
  
 
where
 
where
  
<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/b/b110/b110940/b11094038.png" /></td> </tr></table>
+
$$
 +
F ( x ) = {\mathsf P} \left \{  \sup  _ {0 \leq  t \leq 1 } \eta ( t ) \leq  x \right \} .
 +
$$
  
 
Explicitly,
 
Explicitly,
  
<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/b/b110/b110940/b11094039.png" /></td> </tr></table>
+
$$
 +
F ( x ) = \sum _ {j = - \infty } ^  \infty  ( 1 - 4j  ^ {2} x  ^ {2} ) e ^ {- 2j  ^ {2} x  ^ {2} } =
 +
$$
  
<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/b/b110/b110940/b11094040.png" /></td> </tr></table>
+
$$
 +
=  
 +
{
 +
\frac{\sqrt 2 \pi ^ {5/2 } }{x  ^ {3} }
 +
} \sum _ {j = 0 } ^  \infty  j  ^ {2} e ^ {- { {j  ^ {2} \pi  ^ {2} } / {( 2x  ^ {2} ) } } }
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110940/b11094043.png" />.
+
for $  x > 0 $
 +
and $  F ( x ) = 0 $
 +
for $  x \leq  0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Chung,  "Excursions in Brownian Motion"  ''Arkiv für Math.'' , '''14'''  (1976)  pp. 157–179</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Itô,  H.P. McKean,  "Diffusion processes and their sample paths" , Springer  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Lévy,  "Processus stochastiques et mouvement Brownien" , Gauthier-Villars  (1965)  (Edition: Second)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Takács,  "A Bernoulli excursion and its various applications"  ''Adv. in Probability'' , '''23'''  (1991)  pp. 557–585</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Chung,  "Excursions in Brownian Motion"  ''Arkiv für Math.'' , '''14'''  (1976)  pp. 157–179</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Itô,  H.P. McKean,  "Diffusion processes and their sample paths" , Springer  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Lévy,  "Processus stochastiques et mouvement Brownien" , Gauthier-Villars  (1965)  (Edition: Second)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Takács,  "A Bernoulli excursion and its various applications"  ''Adv. in Probability'' , '''23'''  (1991)  pp. 557–585</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


Brownian excursion process

The limiting process of a Bernoulli excursion. If $ \{ \eta _ {0} , \dots, \eta _ {2n } \} $ is a Bernoulli excursion, and if $ n \rightarrow \infty $, then the finite-dimensional distributions of the process $ \{ { {{\eta _ {[ 2nt ] } } / {\sqrt {2n } } } } : {0 \leq t \leq 1 } \} $ converge to the corresponding finite-dimensional distributions of a process $ \{ {\eta ( t ) } : {0 \leq t \leq 1 } \} $ which is called a Brownian excursion (process). The Brownian excursion process $ \{ {\eta ( t ) } : {0 \leq t \leq 1 } \} $ is a Markov process for which $ {\mathsf P} \{ \eta ( 0 ) = 0 \} = {\mathsf P} \{ \eta ( 1 ) = 0 \} = 1 $ and $ {\mathsf P} \{ \eta ( t ) \geq 0 \} = 1 $ for $ 0 \leq t \leq 1 $. If $ 0 < t < 1 $, then $ \eta ( t ) $ has a density function $ f ( t,x ) $. Obviously, $ f ( t,x ) = 0 $ for $ x \leq 0 $. If $ 0 < t < 1 $ and $ x > 0 $, then

$$ f ( t,x ) = { \frac{2x ^ {2} }{\sqrt {2 \pi t ^ {3} ( 1 - t ) ^ {3} } } } e ^ {- { {x ^ {2} } / {( 2t ( 1 - t ) ) } } } . $$

If $ 0 < t < u < 1 $, then the random variables $ \eta ( t ) $ and $ \eta ( u ) $ have a joint density function $ f ( t,x;u,y ) $. One finds that $ f ( t,x;u,y ) = 0 $ if $ x \leq 0 $ or $ y \leq 0 $. If $ 0 < t < u < 1 $ and $ x > 0 $, $ y > 0 $, then

$$ f ( t,x;u,y ) = $$

$$ = { \frac{\sqrt {8 \pi } xy }{\sqrt {t ^ {3} ( u - t ) ( 1 - u ) ^ {3} } } } \phi \left ( { \frac{x}{\sqrt t } } \right ) \phi \left ( { \frac{y}{\sqrt {1 - u } } } \right ) \times $$

$$ \times \left [ \phi \left ( { \frac{x - y }{\sqrt {u - t } } } \right ) - \phi \left ( { \frac{x + y }{\sqrt {u - t } } } \right ) \right ] , $$

where

$$ \phi ( x ) = { \frac{1}{\sqrt {2 \pi } } } e ^ { {{x ^ {2} } / 2 } } $$

is the normal density function (cf. Normal distribution; Density of a probability distribution). Since $ \{ {\eta ( t ) } : {0 \leq t \leq 1 } \} $ is a Markov process, the density functions $ f ( t,x ) $ and $ f ( t,x;u,y ) $ completely determine its finite-dimensional distributions. For various properties of the Brownian excursion process, see [a3], [a1], [a4].

The distributions of various functionals of the Brownian excursion $ \{ {\eta ( t ) } : {0 \leq t \leq 1 } \} $ frequently appear in probability theory. In particular, many limit distributions of the Bernoulli excursion $ \{ \eta _ {0} \dots \eta _ {2n } \} $ can be expressed simply as the distributions of certain functionals of the Brownian excursion. For example, if $ \delta _ {n} = \max ( \eta _ {0} , \dots, \eta _ {2n } ) $, then

$$ {\lim\limits } _ {n \rightarrow \infty } {\mathsf P} \left \{ { \frac{\delta _ {n} }{\sqrt {2n } } } \leq x \right \} = F ( x ) , $$

where

$$ F ( x ) = {\mathsf P} \left \{ \sup _ {0 \leq t \leq 1 } \eta ( t ) \leq x \right \} . $$

Explicitly,

$$ F ( x ) = \sum _ {j = - \infty } ^ \infty ( 1 - 4j ^ {2} x ^ {2} ) e ^ {- 2j ^ {2} x ^ {2} } = $$

$$ = { \frac{\sqrt 2 \pi ^ {5/2 } }{x ^ {3} } } \sum _ {j = 0 } ^ \infty j ^ {2} e ^ {- { {j ^ {2} \pi ^ {2} } / {( 2x ^ {2} ) } } } $$

for $ x > 0 $ and $ F ( x ) = 0 $ for $ x \leq 0 $.

References

[a1] K.L. Chung, "Excursions in Brownian Motion" Arkiv für Math. , 14 (1976) pp. 157–179
[a2] K. Itô, H.P. McKean, "Diffusion processes and their sample paths" , Springer (1965)
[a3] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) (Edition: Second)
[a4] L. Takács, "A Bernoulli excursion and its various applications" Adv. in Probability , 23 (1991) pp. 557–585
How to Cite This Entry:
Brownian excursion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brownian_excursion&oldid=15228
This article was adapted from an original article by L. Takács (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article