Brownian excursion

Properties

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

${\displaystyle M_{+}\equiv \underset{0\le t\le 1}{sup}e(t)\stackrel{d}{=}\underset{0\le t\le 1}{sup}{W}_0(t)-\underset{0\le t\le 1}{inf}{W}_0(t),}$

(this can also be derived by explicit calculations) and

${\displaystyle {\int }_0^{1}e(t)dt\stackrel{d}{=}{\int }_0^1{W}_0(t)dt-\underset{0\le t\le 1}{inf}{W}_0(t).}$

The following result holds:

${\displaystyle E{M}_{+}=\sqrt{\pi /2}\approx 1.25331\dots ,}$

and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:

${\displaystyle E{M}_{+}^2\approx 1.64493\dots ,\mathrm{Var}(M_{+})\approx 0.0741337\dots .}$

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of ${\displaystyle {\int }_0^{1}e(t)dt}$. 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 ${\displaystyle W}$ in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of ${\displaystyle W}$.

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

With probability 1, a Wiener process is continuous, which means the set on which it is non-zero is an open subset of the real line, thus it is the union of countably many Brownian excursions.

The Brownian excursion area

${\displaystyle A_{+}\equiv {\int }_0^{1}e(t)dt}$

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 ${\displaystyle A_{+}}$ has density

${\displaystyle f_{A_{+}}(x)=\frac{2\sqrt{6}}{x^2}\sum _{j=1}^{\mathrm{\infty }}{v}_j^{2/3}{e}^{-v_j}U\left(-\frac{5}{6},\frac{4}{3};v_j\right)\text{ with }{v}_j=\frac{2|a_j{|}^3}{27x^2}}$

where ${\displaystyle a_j}$ are the zeros of the Airy function and ${\displaystyle U}$ is the confluent hypergeometric function. Janson and Louchard (2007) show that

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

and

${\displaystyle P(A_{+}>x)\sim \frac{6\sqrt{6}}{\sqrt{\pi }}x{e}^{-6x^2}\text{ as }x\to \mathrm{\infty }.}$

They also give higher-order expansions in both cases.

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

${\displaystyle E(A_{+})=\frac{1}{2}\sqrt{\frac{\pi }{2}},E(A_{+}^2)=\frac{5}{12}\approx 0.416666\dots ,\mathrm{Var}(A_{+})=\frac{5}{12}-\frac{\pi }{8}\approx .0239675\dots .}$

Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees.

Related processes

• Brownian bridge
• Brownian meander
• reflected Brownian motion
• skew Brownian motion