Electronic Journal of Probability

Asymptotic properties of expansive Galton-Watson trees

Romain Abraham and Jean-François Delmas

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Abstract

We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 15, 51 pp.

Dates
Received: 12 December 2017
Accepted: 30 January 2019
First available in Project Euclid: 22 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1550826098

Digital Object Identifier
doi:10.1214/19-EJP272

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems

Keywords
conditioned Galton-Watson trees local limits Martin boundary

Rights
Creative Commons Attribution 4.0 International License.

Citation

Abraham, Romain; Delmas, Jean-François. Asymptotic properties of expansive Galton-Watson trees. Electron. J. Probab. 24 (2019), paper no. 15, 51 pp. doi:10.1214/19-EJP272. https://projecteuclid.org/euclid.ejp/1550826098


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