THE COALESCENT STRUCTURE OF GALTON-WATSON TREES IN VARYING ENVIRONMENTS

• 成果类型: Article
• 署名作者: Harris, Simon c.; Palau, Sandra; Pardo, Juan carlos
• 署名单位: University of Auckland; Universidad Nacional Autonoma de Mexico
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2094
• 发表日期: 2024
• 页码: 5388-5425
• 关键词: branching-processes; limit

摘要

We investigate the genealogy of a sample of k >= 2 particles chosen uniformly without replacement from a population alive at large times in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the k - 1 pairwise mergers look like a mixture of independent identically distributed times. Our approach uses k distinguished spine particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement, as required, but additionally (b) there is k-size biasing and discounting according to the population size. Our work significantly extends the spine techniques developed in Harris, Johnston and Roberts (Ann. Appl. Probab. (2020) 30 1368-1414) for genealogies of uniform samples of size k in near-critical continuous-time Galton-Watson processes, as well as a two-spine GWVE construction in Cardona and Palau ( Bernoulli (2021) 27 1643-1665). Our results complement recent works by Kersting ( Proc. Steklov Inst. Maths. (2022) 316 209-219) and Boenkost, Foutel-Rodier and Schertzer (arXiv:2207.11612).