On the Height Profile of a Conditioned Galton-Watson Tree

TL;DR

This paper proves the asymptotic shape of the height profile of conditioned Galton-Watson trees, extending previous results to cases with infinite variance offspring distributions, and showing convergence to a process related to Brownian or Lévy excursions.

Contribution

It provides a new proof for the height profile of Galton-Watson trees, including infinite variance cases, where previous methods based on Brownian local time no longer apply.

Findings

Height profile converges to local time of Brownian excursion for finite variance.

Extended the result to infinite variance offspring distributions.

Developed a new proof strategy for infinite variance cases.

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. L\'{e}vy excursions.