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

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.

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• Branching random walks conditioned on rarely survival
• Maximum likelihood estimation for spinal-structured trees
• The Horton-Strahler number of Galton-Watson trees with possibly infinite variance
• Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations