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

2014/07/17 by Hongwei Bi, Bi, Hongwei, Jean‐François Delmas +2 · 1 citation
Computer Science · Mathematics · #60G55 #92D25 #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary: 60J80 #Probability (math.PR) #Secondary: 60G10 #Stochastic processes and statistical mechanics #math.PR #msc:60G10 #msc:60G55 #msc:60J80 #msc:92D25

paper · pdf · doi:10.48550/arxiv.1407.4539

29 pages

arxiv created 2014/07/17 · openalex publication_date 2014/07/17 · arxiv updated 2014/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of the total length process for the genealogical tree of a population model with random size given by a 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 constant size population associated to the Kingma coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.

Citations

• The conditioned reconstructed process
• Growth of Levy trees

Cited by

• Site Frequency Spectrum in stationary branching populations

Related

• Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length
• Some properties of stationary continuous state branching processes
• Smaller population size at the MRCA time for stationary branching processes
• Continuum random trees and branching processes with immigration
• Asymptotic genealogy of a critical branching process