This thesis concerns applications of some probabilistic tools to phylogeny reconstruction and population genetics. Modelling the evolution of species by continuous-time random walks on the signed permutation groups, we study the asymptotic medians of a set of random permutations sampled from simple random walks at time 0.25cn, for c> 0. Running k independent random walks all starting at identity, we prove that the medians approximate the ancestor (identity permutation) up to time 0.25n, while there exists a constant c>1 after which the medians loose credibility as an estimator. We study the median of a set of random permutations on the symmetric group endowed with different metrics. In particular, for a special metric of dissimilarity, called breakpoint, where the space is not geodesic, we find a large group of medians of random permutations using the concept of partial geodesics (or geodesic patches). Also, we study the Fleming-Viot process in random environment (FVRE) via martingale and duality methods. We develop the duality method to the case of time-dependent and quenched martingale problems. Using a family of dual processes we prove the convergence of the Moran processes in random environments to FVRE in Skorokhod topology. We also study the long-time behaviour of FVRE and prove the existence of equilibrium for the joint annealed-environment process and prove an ergodic theorem for the latter.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/34623 |
| Date | January 2016 |
| Creators | Jamshidpey, Arash |
| Contributors | Sankoff, David, Dawson, Donald A. |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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