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

Abstract

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 Kingman 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.

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

“…As part of Theorem 5.1, we also get that L t coincides with the limit of the shifted total length L ε of the genealogical tree up to t − ε of the individuals alive at time t obtained in [11]: the sequence (L ε − E[L ε |Z t ], ε > 0) converges a.s. towards L t as ε goes down to zero. Intuitively, taking ε n (random) such t − ε n is the first time backward where the number of ancestors of the extant population at time t is n, we get that L ε n is distributed as t−ε n ,n .…”

“…The static simulation of Subsection 4.1 allows us to give precise asymptotics for the first and second moments (conditionally given the population size) of the quantity n . Then we prove that n − L ε (where L ε is the length of the genealogical tree up to level −ε introduced in [11]) converges in L 2 to 0 when ε is of order 1/n, which easily yields the theorem.…”

“…We refer to [11] for properties of the stationary process (L t , t ∈ R). We only recall that the Laplace transform of L t is given, for λ > 0, by…”

Coalescences in continuous-state branching processes

“…Starting from a different point of view, Bi and Delmas [BD16] and Chen and Delmas [CD12] have considered stationary subcritical branching population obtained as processes conditioned on the non-extinction. The genealogy is then studied via a Poisson representation of the population.…”

Reversal property of the Brownian tree

“…For s < t, let M t s be the number of individuals at time s who have descendants at time t. It is proven in Bi and Delmas [5], that for fixed θ > 0 a time reversal property holds: in the stationary regime, the ancestor process ((M s s−r , r > 0), s ∈ R) is distributed as the descendant process ((M s+r s , r > 0), s ∈ R), see Remark 3.14. This paper extends and explains this identity in law by reversing the genealogical tree.…”