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January 2010 The contour of splitting trees is a Lévy process
Amaury Lambert
Ann. Probab. 38(1): 348-395 (January 2010). DOI: 10.1214/09-AOP485

## Abstract

Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure.

A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (v, τ) of some individual v (vertex) in a discrete tree where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping φ from the tree into the real line which preserves this order. The inverse of φ is called the exploration process, and the projection of this inverse on chronological levels the contour process.

For splitting trees truncated up to level τ, we prove that a thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

## Citation

Amaury Lambert. "The contour of splitting trees is a Lévy process." Ann. Probab. 38 (1) 348 - 395, January 2010. https://doi.org/10.1214/09-AOP485

## Information

Published: January 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1190.60083
MathSciNet: MR2599603
Digital Object Identifier: 10.1214/09-AOP485

Subjects:
Primary: 60J80
Secondary: 37E25 , 60G51 , 60G55 , 60G70 , 60J55 , 60J75 , 60J85 , 92D25

Keywords: Coalescent point process , composition of subordinators , contour process , Crump–Mode–Jagers branching process , Exploration process , Jirina process , Lévy process , limit theorems , Malthusian parameter , modified geometric distribution , Poisson point process , Population dynamics , Real trees , scale function , Yaglom distribution

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Vol.38 • No. 1 • January 2010