Probability Surveys

Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas

Svante Janson

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Abstract

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright’s constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area.

The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.

Article information

Source
Probab. Surveys, Volume 4 (2007), 80-145.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.ps/1178804352

Digital Object Identifier
doi:10.1214/07-PS104

Mathematical Reviews number (MathSciNet)
MR2318402

Zentralblatt MATH identifier
1189.60147

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 05C30: Enumeration in graph theory 60G15: Gaussian processes

Citation

Janson, Svante. Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas. Probab. Surveys 4 (2007), 80--145. doi:10.1214/07-PS104. https://projecteuclid.org/euclid.ps/1178804352


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