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Annals of Combinatorics

, Volume 12, Issue 4, pp 403–416 | Cite as

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

  • James Allen FillEmail author
  • Svante Janson
Article

Abstract

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown.

Keywords

large deviations tail asymptotics Galton-Watson trees simply generated families of trees Brownian excursion variational problems total path length Wiener index 

AMS Subject Classification

60F10 60C05 60J65 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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