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.[1]

Definition

A Brownian excursion process, ${\displaystyle e}$, 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 ${\displaystyle e}$ 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 ${\displaystyle \tau _{-}}$ that W hits zero before time 1 and the first time ${\displaystyle \tau _{+}}$ that Brownian motion ${\displaystyle W}$ hits zero after time 1:[2]

${\displaystyle \{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{{\frac {|W((1-t)\tau _{-}+t\tau _{+})|}{\sqrt {\tau _{+}-\tau _{-}}}}:\ 0\leq t\leq 1\right\}.}$

Let ${\displaystyle \tau _{m}}$ be the time that a Brownian bridge process ${\displaystyle W_{0}}$ achieves its minimum on [0, 1]. Vervaat (1979) shows that

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

Properties

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

${\displaystyle M_{+}\equiv \sup _{0\leq t\leq 1}e(t)\ {\stackrel {d}{=}}\ \sup _{0\leq t\leq 1}W_{0}(t)-\inf _{0\leq t\leq 1}W_{0}(t),}$

(this can also be derived by explicit calculations[3][4]) and

${\displaystyle \int _{0}^{1}e(t)\,dt\ {\stackrel {d}{=}}\ \int _{0}^{1}W_{0}(t)\,dt-\inf _{0\leq t\leq 1}W_{0}(t).}$

The following result holds:[5]

${\displaystyle EM_{+}={\sqrt {\pi /2}}\approx 1.25331\ldots ,\,}$

and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:[5]

${\displaystyle EM_{+}^{2}\approx 1.64493\ldots \ ,\ \ \operatorname {Var} (M_{+})\approx 0.0741337\ldots .}$

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of ${\displaystyle \int _{0}^{1}e(t)\,dt}$. 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

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.[6][7][8][9][10] and the limit distribution of the Betti numbers of certain varieties in cohomology theory.[11] Takacs (1991a) shows that ${\displaystyle A_{+}}$ has density

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

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

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

and

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

They also give higher-order expansions in both cases.

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

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

Brownian excursions also arise in connection with queuing problems,[12] railway traffic,[13][14] and the heights of random rooted binary trees.[15]

Notes

1. Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)
2. Itô and McKean (1974, page 75)
3. Chung (1976)
4. Kennedy (1976)
5. Durrett and Iglehart (1977)
6. Wright, E. M. (1977). "The number of connected sparsely edged graphs". Journal of Graph Theory. 1 (4): 317–330. doi:10.1002/jgt.3190010407.
7. Wright, E. M. (1980). "The number of connected sparsely edged graphs. III. Asymptotic results". Journal of Graph Theory. 4 (4): 393–407. doi:10.1002/jgt.3190040409.
8. Spencer J (1997). "Enumerating graphs and Brownian motion". Communications on Pure and Applied Mathematics. 50 (3): 291–294. doi:10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6.
9. Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys. 4: 80–145. arXiv:0704.2289. Bibcode:2007arXiv0704.2289J. doi:10.1214/07-PS104.
10. Flajolet, P.; Louchard, G. (2001). "Analytic variations on the Airy distribution". Algorithmica. 31 (3): 361–377. CiteSeerX 10.1.1.27.3450. doi:10.1007/s00453-001-0056-0.
11. Reineke M (2005). "Cohomology of noncommutative Hilbert schemes". Algebras and Representation Theory. 8 (4): 541–561. arXiv:math/0306185. doi:10.1007/s10468-005-8762-y.
12. Iglehart D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive". The Annals of Probability. 2 (4): 608–619. doi:10.1214/aop/1176996607.
13. Takacs L (1991a). "A Bernoulli excursion and its various applications". Advances in Applied Probability. 23 (3): 557–585. doi:10.1017/s0001867800023739.
14. Takacs L (1991b). "On a probability problem connected with railway traffic". Journal of Applied Mathematics and Stochastic Analysis. 4: 263–292. doi:10.1155/S1048953391000011.
15. Takacs L (1994). "On the Total Heights of Random Rooted Binary Trees". Journal of Combinatorial Theory, Series B. 61 (2): 155–166. doi:10.1006/jctb.1994.1041.