The Annals of Probability

The contour of splitting trees is a Lévy process

Amaury Lambert

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.

Article information

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

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1264434002

Digital Object Identifier
doi:10.1214/09-AOP485

Mathematical Reviews number (MathSciNet)
MR2599603

Zentralblatt MATH identifier
1190.60083

Citation

Lambert, Amaury. The contour of splitting trees is a Lévy process. Ann. Probab. 38 (2010), no. 1, 348--395. doi:10.1214/09-AOP485. https://projecteuclid.org/euclid.aop/1264434002