Local limits of conditioned Galton-Watson trees: the infinite spine case. (English) Zbl 1285.60085

Abraham, Romain; Delmas, Jean-François

Electron. J. Probab. 19, Paper No. 2, 19 p. (2014).

Summary: 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 the 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.

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60B10	Convergence of probability measures

Keywords:

Galton-Watson; random tree; local-limit; non-extinction; branching process