2022/07/22 by Harris Simon C., Harris, Simon C., Palau Sandra +2
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F17 #60G09 #60J80 #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2207.10923
openalex publication_date 2022/07/22 · openalex created_date 2022/07/27 · openalex updated_date 2026/07/28
We investigate the genealogy of a sample of k≥1 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 [Annals Applied Probability, 2020] 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]. Our results complement recent works by Kersting [Proc. Steklov Inst. Maths., 2022] and Boenkost, Foutel-Rodier, and Schertzer [arXiv:2207.11612].