Brownian excursion

In probability theory a Brownian excursion process (BPE) 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]

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. Although note that, since the probability for an unrestricted Brownian bridge to be positive is zero, the conditioning requires care.

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]

${\displaystyle \{e(t):0\le t\le 1\}\stackrel{d}{=}\{\frac{|W((1-t){\tau }_{-}+t{\tau }_{+})|}{\sqrt{{\tau }_{+}-{\tau }_{-}}}:0\le t\le 1\},}$

where the square root is due to the square root self-similarity of Wiener process. That is,

is a Wiener process for any fixed constant

.

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

${\displaystyle \{e(t):0\le t\le 1\}\stackrel{d}{=}\{W_0({\tau }_m+tmod1)-W_0({\tau }_m):0\le t\le 1\}.}$

This is sometimes called Vervaat's transformation.

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^[3][4]) and

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

The following result holds:^[5]

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

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

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

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

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

${\displaystyle A_{+}\equiv {\displaystyle \underset{0}{\overset{1}{\int }}}e(t)dt}$

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

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

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,^[12] railway traffic,^[13][14] and the heights of random rooted binary trees.^[15]

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

• 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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