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 particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. 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.

Definition

A Brownian excursion process, $e$ , 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.

Another representation of a Brownian excursion $e$ in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyoshi Itō and Henry P. McKean, Jr.) is in terms of the last time $\tau _{-}$ that W hits zero before time 1 and the first time $\tau _{+}$ that Brownian motion $W$ hits zero after time 1:

$\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{{\frac {|W((1-t)\tau _{-}+t\tau _{+})|}{\sqrt {\tau _{+}-\tau _{-}}}}:\ 0\leq t\leq 1\right\}.$ Let $\tau _{m}$ be the time that a Brownian bridge process $W_{0}$ achieves its minimum on [0, 1]. Vervaat (1979) shows that

$\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{W_{0}(\tau _{m}+t{\bmod {1}})-W_{0}(\tau _{m}):\ 0\leq t\leq 1\right\}.$ Properties

Vervaat's representation of a Brownian excursion has several consequences for various functions of $e$ . In particular:

$M_{+}\equiv \sup _{0\leq t\leq 1}e(t)\ {\stackrel {d}{=}}\ \sup _{0\leq t\leq 1}W_{0}(t)-\inf _{0\leq t\leq 1}W_{0}(t),$ (this can also be derived by explicit calculations) and

$\int _{0}^{1}e(t)\,dt\ {\stackrel {d}{=}}\ \int _{0}^{1}W_{0}(t)\,dt-\inf _{0\leq t\leq 1}W_{0}(t).$ The following result holds:

$EM_{+}={\sqrt {\pi /2}}\approx 1.25331\ldots ,\,$ and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:

$EM_{+}^{2}\approx 1.64493\ldots \ ,\ \ \operatorname {Var} (M_{+})\approx 0.0741337\ldots .$ Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of $\int _{0}^{1}e(t)\,dt$ . 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 $W$ in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of $W$ .

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.

Connections and applications

The Brownian excursion area

$A_{+}\equiv \int _{0}^{1}e(t)\,dt$ 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 $A_{+}$ has density

$f_{A_{+}}(x)={\frac {2{\sqrt {6}}}{x^{2}}}\sum _{j=1}^{\infty }v_{j}^{2/3}e^{-v_{j}}U\left(-{\frac {5}{6}},{\frac {4}{3}};v_{j}\right)\ \ {\text{ with }}\ \ v_{j}={\frac {2|a_{j}|^{3}}{27x^{2}}}$ where $a_{j}$ are the zeros of the Airy function and $U$ is the confluent hypergeometric function. Janson and Louchard (2007) show that

$f_{A_{+}}(x)\sim {\frac {72{\sqrt {6}}}{\sqrt {\pi }}}x^{2}e^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty ,$ and

$P(A_{+}>x)\sim {\frac {6{\sqrt {6}}}{\sqrt {\pi }}}xe^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty .$ They also give higher-order expansions in both cases.

Janson (2007) gives moments of $A_{+}$ and many other area functionals. In particular,

$E(A_{+})={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}},\ \ E(A_{+}^{2})={\frac {5}{12}}\approx 0.416666\ldots ,\ \ \operatorname {Var} (A_{+})={\frac {5}{12}}-{\frac {\pi }{8}}\approx .0239675\ldots \ .$ Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees.