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