Brownian excursion

Alternatively, it is a Brownian bridge process conditioned to be positive.

BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.

, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1.

Alternatively, it is a Brownian bridge process conditioned to be positive.

in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.[2]) is in terms of the last time

In particular: (this can also be derived by explicit calculations[3][4]) and The following result holds:[5] and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:[5] Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of

A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984).

Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion

in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of

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.[6][7][8][9][10] and the limit distribution of the Betti numbers of certain varieties in cohomology theory.

Janson and Louchard (2007) show that and They also give higher-order expansions in both cases.

In particular, Brownian excursions also arise in connection with queuing problems,[12] railway traffic,[13][14] and the heights of random rooted binary trees.