Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Total length of the genealogical tree for quadratic stationary continuous-state branching processes

Hongwei Bi and Jean-François Delmas

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Abstract

We prove the existence of the total length process for the genealogical tree of a population model with random size given by quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for a constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all times, and a description of the so-called lineage tree in this model.

Résumé

Nous démonstrons l’existence du processus de longueur totale renormalisée pour l’arbre généalogique dans un modèle de population dont la taille évolue suivant un processus de branchement continu quadratique (diffusion de Feller). Nous donnons également la loi unidimensionnelle de la longueur totale de l’arbre généalogique ainsi que les fluctuations associées à la renormalisation. Ce résultat est à rapprocher de ceux obtenus par Pfaffelhuber et Wakolbinger dans le cadre d’une population de taille constante associée au processus de coalescence de Kingman. Nous établissons également une propriété d’invariance par retournement du temps pour le processus du nombre des ancêtres qui permet d’obtenir en particulier une description du processus ancestral dans ce modèle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1321-1350.

Dates
Received: 18 July 2014
Revised: 16 April 2015
Accepted: 16 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723522

Digital Object Identifier
doi:10.1214/15-AIHP683

Mathematical Reviews number (MathSciNet)
MR3531711

Zentralblatt MATH identifier
1350.60090

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D25: Population dynamics (general)
Secondary: 60G10: Stationary processes 60G55: Point processes

Keywords
Branching process Population model Genealogical tree Lineage tree Time-reversal

Citation

Bi, Hongwei; Delmas, Jean-François. Total length of the genealogical tree for quadratic stationary continuous-state branching processes. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1321--1350. doi:10.1214/15-AIHP683. https://projecteuclid.org/euclid.aihp/1469723522


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