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The contour of splitting trees is a Lévy process. (English) Zbl 1190.60083
Author’s 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 ${\infty }$). 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 $(v, \tau )$ of some individual $v$ (vertex) in a discrete tree where $\tau $ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping $\varphi $ from the tree into the real line which preserves this order. The inverse of $\varphi $ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level $\tau $, we prove that a thus defined contour process is a Lévy process reflected below $\tau $ 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.

MSC:
60J80Branching processes
37E25Maps of trees and graphs
60G51Processes with independent increments; Lévy processes
60G55Point processes
60G70Extreme value theory; extremal processes (probability theory)
60J55Local time, additive functionals
60J75Jump processes
60J85Applications of branching processes
92D25Population dynamics (general)
WorldCat.org
Full Text: DOI
References:
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