On the Height Profile of a Conditioned Galton-Watson Tree

2011/01/19 by Götz Kersting, Kersting, Götz · 1 citation
Computer Science · Mathematics · #60F17 #60J80 #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1101.3656

openalex publication_date 2011/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Drmota and Gittenberger (1997) proved a conjecture due to Aldous (1991) on the height profile of a Galton-Watson tree with an offspring distribution of finite variance, conditioned on a total size of n individuals. The conjecture states that in distribution its shape, more precisely its scaled height profile coincides asymptotically with the local time process of a Brownian excursion of duration 1. We give a proof of the result, which extends to the case of an infinite variance offspring distribution. This requires a different strategy, since in the infinite variance case there is no longer a relationship to the local time of Brownian resp. Lévy excursions.

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