Exact simulation of the genealogical tree for a stationary branching population and application to the asymptotics of its total length

TL;DR

This paper presents an exact simulation method for the genealogical tree of a stationary population modeled by a continuous state branching process with immigration, and analyzes the asymptotic behavior of its total length.

Contribution

It introduces a novel exact simulation procedure for the genealogical tree in a stationary population with immigration and proves the convergence of its total length as the sample size grows.

Findings

Exact simulation procedure for genealogical trees

Convergence of the renormalized total length as sample size increases

Connection to previous models and asymptotic analysis

Abstract

We consider a model of stationary population with random size given by a continuous state branching process with immigration with a quadratic branching mechanism. We give an exact elementary simulation procedure of the genealogical tree of $n$ individuals randomly chosen among the extant population at a given time. Then, we prove the convergence of the renormalized total length of this genealogical tree as $n$ goes to infinity, see also Pfaffelhuber, Wakolbinger and Weisshaupt (2011) in the context of a constant size population. The limit appears already in Bi and Delmas (2016) but with a different approximation of the full genealogical tree. The proof is based on the ancestral process of the extant population at a fixed time which was defined by Aldous and Popovic (2005) in the critical case.