Some properties of stationary continuous state branching processes

2020/05/20 by Abraham, Romain, Jean‐François Delmas, Delmas, Jean-François +2
Business, Management and Accounting · Computer Science · Mathematics · #Advanced Queuing Theory Analysis #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2005.09924

openalex publication_date 2020/05/20 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28

Abstract

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.

Citations

• Sampling theory for neutral alleles in a varying environment
• Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

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