The Annals of Probability

The contour of splitting trees is a Lévy process

Amaury Lambert

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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.

Article information

Ann. Probab. Volume 38, Number 1 (2010), 348-395.

First available in Project Euclid: 25 January 2010

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Mathematical Reviews number (MathSciNet)

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 37E25: Maps of trees and graphs 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60G70: Extreme value theory; extremal processes 60J55: Local time and additive functionals 60J75: Jump processes 60J85: Applications of branching processes [See also 92Dxx] 92D25: Population dynamics (general)

Real trees population dynamics contour process exploration process Poisson point process Crump–Mode–Jagers branching process Malthusian parameter Lévy process scale function composition of subordinators Jirina process coalescent point process limit theorems Yaglom distribution modified geometric distribution


Lambert, Amaury. The contour of splitting trees is a Lévy process. The Annals of Probability 38 (2010), no. 1, 348--395. doi:10.1214/09-AOP485.

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