Brownian excursion the Airy function and formula_22 is the confluent hypergeometric function. Janson and Louchard (2007) show that and They also give higher-order expansions in both cases. Janson (2007) gives moments of formula_19 and many other area functionals. In particular, Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees. Brownian excursion In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In
Brownian excursion Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion formula_5 in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of formula_5. For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII. The Brownian excursion area arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.,, and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that formula_19 has density where formula_21 are the zeros of