Electronic Journal of Probability

Local limits of Markov branching trees and their volume growth

Camille Pagnard

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Abstract

We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed.

We prove that under some natural assumption on this sequence of probabilities, when their sizes go to infinity, the trees converge in distribution to an infinite tree which also satisfies the Markov branching property. Furthermore, when this infinite tree has a single path from the root to infinity, we give conditions to ensure its convergence in distribution under appropriate rescaling of its distance and counting measure to a self-similar fragmentation tree with immigration. In particular, this allows us to determine how, in this infinite tree, the “volume” of the ball of radius $R$ centred at the root asymptotically grows with $R$.

Our unified approach will allow us to develop various new applications, in particular to different models of growing trees and cut-trees, and to recover known results. An illustrative example lies in the study of Galton-Watson trees: the distribution of a critical Galton-Watson tree conditioned on its size converges to that of Kesten’s tree when the size grows to infinity. If furthermore, the offspring distribution has finite variance, under adequate rescaling, Kesten’s tree converges to Aldous’ self-similar CRT and the total size of the $R$ first generations asymptotically behaves like $R^2$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 95, 53 pp.

Dates
Received: 26 August 2016
Accepted: 19 August 2017
First available in Project Euclid: 8 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1510110478

Digital Object Identifier
doi:10.1214/17-EJP96

Mathematical Reviews number (MathSciNet)
MR3724563

Zentralblatt MATH identifier
1378.60112

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

Keywords
random trees Markov branching trees local limits scaling limits self-similar fragmentation trees GHP topology Galton-Watson trees

Rights
Creative Commons Attribution 4.0 International License.

Citation

Pagnard, Camille. Local limits of Markov branching trees and their volume growth. Electron. J. Probab. 22 (2017), paper no. 95, 53 pp. doi:10.1214/17-EJP96. https://projecteuclid.org/euclid.ejp/1510110478


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