Local limits of conditioned Galton-Watson trees I: the infinite spine case

TL;DR

This paper establishes a precise criterion for when conditioned Galton-Watson trees converge to Kesten's tree, providing simpler proofs of known results and new insights into tree limits conditioned on out-degree distributions.

Contribution

It introduces a necessary and sufficient condition for convergence to Kesten's tree and applies it to derive new limit results for trees conditioned on out-degree counts.

Findings

Condition for convergence to Kesten's tree established

Elementary proofs of known local limit results provided

New limit theorems for trees conditioned on out-degree in critical/sub-critical cases

Abstract

We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.