Brownian excursion

Short description: Stochastic process

A realization of 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]

## Contents

* 1 Definition
* 2 Properties
* 3 Connections and applications
* 4 Related processes
* 5 Notes
* 6 References

## Definition

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

[math]\displaystyle{ \\{ e(t) : \ {0 \le t \le 1} \\} \ \stackrel{d}{=} \ \left \\{ \frac{|W((1-t) \tau_{-} + t \tau_{+} )|}{\sqrt{\tau_+ - \tau_{-}}} : \ 0 \le t \le 1 \right \\} . }[/math]

Let [math]\displaystyle{ \tau_m }[/math] be the time that a Brownian bridge process [math]\displaystyle{ W_0 }[/math] achieves its minimum on [0, 1]. Vervaat (1979) shows that

[math]\displaystyle{ \\{ e(t) : \ {0\le t \le 1} \\} \ \stackrel{d}{=} \ \left \\{ W_0 ( \tau_m + t \bmod 1) - W_0 (\tau_m ): \ 0 \le t \le 1 \right \\} . }[/math]

## Properties

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

[math]\displaystyle{ M_{+} \equiv \sup_{0 \le t \le 1} e(t) \ \stackrel{d}{=} \ \sup_{0 \le t \le 1} W_0 (t) - \inf_{0 \le t \le 1} W_0 (t) , }[/math]

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

[math]\displaystyle{ \int_0^1 e(t) \, dt \ \stackrel{d}{=} \ \int_0^1 W_0 (t) \, dt - \inf_{0 \le t \le 1} W_0 (t) . }[/math]

The following result holds:[5]

[math]\displaystyle{ E M_+ = \sqrt{\pi/2} \approx 1.25331 \ldots, \, }[/math]

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

[math]\displaystyle{ E M_+^2 \approx 1.64493 \ldots \ , \ \ \operatorname{Var}(M_+) \approx 0.0741337 \ldots. }[/math]

Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of [math]\displaystyle{ \int_0^1 e(t) \, dt }[/math]. 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 [math]\displaystyle{ W }[/math] in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of [math]\displaystyle{ W }[/math].

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

[math]\displaystyle{ A_+ \equiv \int_0^1 e(t) \, dt }[/math]

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 [math]\displaystyle{ A_+ }[/math] has density

[math]\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} }[/math]

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

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

and

[math]\displaystyle{ P(A_+ \gt x) \sim \frac{6 \sqrt{6}}{\sqrt{\pi}} x e^{- 6x^2} \ \ \text{ as } \ \ x \rightarrow \infty. }[/math]

They also give higher-order expansions in both cases.

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

[math]\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 \ . }[/math]

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

## Related processes

* Brownian bridge
* Brownian meander
* reflected Brownian motion
* skew Brownian motion

## Notes

1. ↑ Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)
2. ↑ 2.0 2.1 Itô and McKean (1974, page 75)
3. ↑ Chung (1976)
4. ↑ Kennedy (1976)
5. ↑ 5.0 5.1 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. doi:10.1214/07-PS104. Bibcode: 2007arXiv0704.2289J.
10. ↑ Flajolet, P.; Louchard, G. (2001). "Analytic variations on the Airy distribution". Algorithmica 31 (3): 361–377. doi:10.1007/s00453-001-0056-0.
11. ↑ Reineke M (2005). "Cohomology of noncommutative Hilbert schemes". Algebras and Representation Theory 8 (4): 541–561. 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.

## References

* Chung, K. L. (1975). "Maxima in Brownian excursions". Bulletin of the American Mathematical Society 81 (4): 742-745. doi:10.1090/s0002-9904-1975-13852-3. http://projecteuclid.org/euclid.bams/1183537153.
* Chung, K. L. (1976). "Excursions in Brownian motion". Arkiv för Matematik 14 (1): 155-177. doi:10.1007/bf02385832. Bibcode: 1976ArM....14..155C.
* Durrett, Richard T.; Iglehart, Donald L. (1977). "Functionals of Brownian meander and Brownian excursion". Annals of Probability 5 (1): 130-135. doi:10.1214/aop/1176995896.
* Groeneboom, Piet (1983). "The concave majorant of Brownian motion". Annals of Probability 11 (4): 1016-1027. doi:10.1214/aop/1176993450.
* Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields 81: 79-109. doi:10.1007/BF00343738. https://ir.cwi.nl/pub/6435.
* Diffusion Processes and their Sample Paths. Classics in Mathematics (Second printing, corrected ed.). Springer-Verlag, Berlin. 2013. ISBN 978-3540606291.
* Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys 4: 80-145. doi:10.1214/07-ps104. Bibcode: 2007arXiv0704.2289J.
* Janson, Svante; Louchard, Guy (2007). "Tail estimates for the Brownian excursion area and other Brownian areas.". Electronic Journal of Probability 12: 1600-1632. doi:10.1214/ejp.v12-471. Bibcode: 2007arXiv0707.0991J. http://www.emis.de/journals/EJP-ECP/article/view/471.html.
* Kennedy, Douglas P. (1976). "The distribution of the maximum Brownian excursion". Journal of Applied Probability 13 (2): 371-376. doi:10.2307/3212843.
* Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. 1948.
* Louchard, G. (1984). "Kac's formula, Levy's local time and Brownian excursion". Journal of Applied Probability 21 (3): 479-499. doi:10.2307/3213611.
* Pitman, J. W. (1983). "Remarks on the convex minorant of Brownian motion". 5. Birkhauser, Boston. pp. 219-227.
* Revuz, Daniel; Yor, Marc (2004). Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. 293. Springer-Verlag, Berlin. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3.
* Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability 7 (1): 143-149. doi:10.1214/aop/1176995155.

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