# Asymptotics, weak convergence and duality in population genetics

**Time: **
Fri 2021-01-08 13.00

**Location: **
Via Zoom https://kth-se.zoom.us/webinar/register/WN_FIDOWUMsQOyhrUSCPfzUUQ, (English)

**Doctoral student: **
Martina Favero
, Matematisk statistik

**Opponent: **
Prof. Alison Etheridge, Oxford University, Oxford, England

**Supervisor: **
Professor Henrik Hult, Matematisk statistik

## Abstract

This thesis consists of four papers on asymptotic results and stochastic duality for some processes in mathematical population genetics. The focus is on Wright-Fisher diffusions and coalescent processes, which model, respectively, the evolution of frequencies of genetic types and genealogies in a population,and play a key role in inference on genetic data sets.

Paper A concerns the derivation of a stochastic dual process for the coupled Wright-Fisher diffusionin a multi-locus population evolving under pairwise selection, which acts on pairs of loci, parent dependent mutations and free recombination. The dual process consists of block counting processes of coupled ancestral selection graphs, one for each locus. In the dual process, coalescence, mutation and single-branching events occur at one locus at a time, whereas double-branching events consist of two branching events occurring at two loci simultaneously.

The remaining three papers provide asymptotic results concerning sequences related to the Kingman coalescent, with parent dependent mutations, as the size of the initial configuration grows to infinity.

In Paper B, it is shown that the sampling probabilities of the Kingman coalescent decay polynomially. The degree of the polynomial depends of the number of types in the model, and the multiplicative constant on the stationary density of the Wright-Fisher diffusion.Moreover, the asymptotic behaviour of the backward transition probabilities is analysed.

In Paper C, it is proved that the normalised and time scaled jump chains of the block counting process of the Kingman coalescent weakly converge to a deterministic process. Furthermore, the time scaled chains counting the number of mutations between types weakly converge to independent Poisson processes with varying intensities.

Paper D focuses on establishing a theoretical framework for the asymptotic analysis of importance sampling algorithms for the coalescent. To this aim, the weak convergence result in Paper C is extended to include a cost component. It is illustrated how the weak convergence of the cost sequence can be used to study the asymptotic behaviour of importance sampling weights, and consequently asymptotic properties of the corresponding algorithms.