# Brownian excursion

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=46167
This article was adapted from an original article by L. TakÃ¡cs (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article