Precise logarithmic asymptotics for the right tails of some limit random variables for random trees

2007/01/09 by James Allen Fill, Svante Janson, Fill, James Allen +1
Decision Sciences · Mathematics · #60C05 #60F10 #60J65 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/0701259

openalex publication_date 2007/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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