Abstract
Take a continuoustime GaltonWatson tree. If the system survives until a large time T , then choose k particles uniformly from those alive. What does the ancestral tree drawn out by these k particles look like? Some special cases are known but we give a more complete answer.We concentrate on nearcritical cases where the mean number of offspring is 1 + μ/T for some μ ∈ R, and show that a scaling limit exists as T →∞. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but the times of coalescence have an interesting and highly nontrivial structure. The randomly fluctuating population size, as opposed to constant size populations where the Kingman coalescent more usually arises, have a pronounced effect on both the results and the method of proof required. We give explicit formulas for the distribution of the coalescent times, as well as a construction of the genealogical tree involving a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birthdeath processes.
Original language  English 

Pages (fromto)  13681414 
Number of pages  47 
Journal  Annals of Applied Probability 
Volume  30 
Issue number  3 
Early online date  29 Jul 2020 
DOIs  
Publication status  Published  2020 
Bibliographical note
Funding Information:M. I. Roberts was supported during the early stages of this work by EPSRC fellowship EP/K007440/1, and during the latter stages by a Royal Society University Research Fellowship. S. G. G. Johnston was supported during for the major part of this work by University of Bath URS funding.
Publisher Copyright:
© Institute of Mathematical Statistics, 2020.
Keywords
 Coalescent
 GaltonWatson tree
 Genealogy
 Spine
ASJC Scopus subject areas
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Matthew Roberts
 Department of Mathematical Sciences  Royal Society University Research Fellow
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching