## Electronic Journal of Probability

### Local limits of conditioned Galton-Watson trees: 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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 2, 19 pp.

Dates
Accepted: 3 January 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065644

Digital Object Identifier
doi:10.1214/EJP.v19-2747

Mathematical Reviews number (MathSciNet)
MR3164755

Zentralblatt MATH identifier
1285.60085

Rights

#### Citation

Abraham, Romain; Delmas, Jean-François. Local limits of conditioned Galton-Watson trees: the infinite spine case. Electron. J. Probab. 19 (2014), paper no. 2, 19 pp. doi:10.1214/EJP.v19-2747. https://projecteuclid.org/euclid.ejp/1465065644

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