THE CONTOUR OF SPLITTING TREES IS A LÉVY PROCESS
By Amaury Lambert. 2007
125 citations recorded by the source corpus.
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 newboms 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 (ν, τ) of some individual ν (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 r, we prove that a thus defined contour process is a Levy 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.
Subfields: Condensed Matter Physics, Management Science and Operations Research, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0de1a911dae15b4e1e083c5fb3e20e11511eaf54
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In Condensed Matter Physics
In Management Science and Operations Research
In Mathematical Physics
Citations in the corpus
- The coalescent structure of continuous-time Galton–Watson trees (2017, methodology): “[17] Lambert, A. (2010).”
- Population Dynamics and Random Genealogies (2008, unknown): “[82,83].”