Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Clément Foucart and Martin Möhle

Stochastic Processes and their Applications, 2022, vol. 150, issue C, 510-531

Abstract: Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). We study the flow of ancestral lineages as time goes to the past and show that, properly renormalized, it converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. The inverse subordinator is shown to be partitioning the current population into ancestral families with distinct common ancestors. When Grey’s condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families have i.i.d. sizes distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey’s condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points S of the limiting inverse subordinator. The proof is based on a general result for stochastically monotone processes of independent interest, which relates θ-invariant measures and θ-invariant functions for a process and its Siegmund dual.

Keywords: Branching processes; Continuous-state space; Inverse subordinators; Ancestral lineage; Siegmund dual; Invariant function (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414922001090
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:150:y:2022:i:c:p:510-531

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2022.05.001

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().