Brownian excursion

Brownian excursion process

The limiting process of a Bernoulli excursion. If is a Bernoulli excursion, and if , then the finite-dimensional distributions of the process converge to the corresponding finite-dimensional distributions of a process which is called a Brownian excursion (process). The Brownian excursion process is a Markov process for which and for . If , then has a density function . Obviously, for . If and , then

If , then the random variables and have a joint density function . One finds that if or . If and , , then

where

is the normal density function (cf. Normal distribution; Density of a probability distribution). Since is a Markov process, the density functions and 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 frequently appear in probability theory. In particular, many limit distributions of the Bernoulli excursion can be expressed simply as the distributions of certain functionals of the Brownian excursion. For example, if , then

where

Explicitly,

for and for .

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