# Genealogical Properties of Subsamples in Highly Fecund Populations

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## Abstract

We consider some genealogical properties of nested samples. The complete sample is assumed to have been drawn from a natural population characterised by high fecundity and sweepstakes reproduction (abbreviated HFSR). The random gene genealogies of the samples are—due to our assumption of HFSR—modelled by coalescent processes which admit multiple mergers of ancestral lineages looking back in time. Among the genealogical properties we consider are the probability that the most recent common ancestor is shared between the complete sample and the subsample nested within the complete sample; we also compare the lengths of ‘internal’ branches of nested genealogies between different coalescent processes. The results indicate how ‘informative’ a subsample is about the properties of the larger complete sample, how much information is gained by increasing the sample size, and how the ‘informativeness’ of the subsample varies between different coalescent processes.

## Keywords

Coalescent High fecundity Nested samples Multiple mergers Time to most recent common ancestor## Mathematics Subject Classification

92D15 60J28## Notes

### Acknowledgements

We thank Alison Etheridge for many and very valuable comments and suggestions, especially regarding Theorem 1. BE was funded by DFG grant STE 325/17-1 to Wolfgang Stephan through Priority Programme SPP1819: Rapid Evolutionary Adaptation. FF was funded by DFG grant FR 3633/2-1 through Priority Program 1590: Probabilistic Structures in Evolution.

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