# Probability Workshops

**TRINITY TERM 2017**

Probability Workshops are held on **Mondays from 12:00 - 1:15 pm** during term time (unless indicated otherwise) in L4 at The Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter.

**Week 1 - Monday 24th April**

**Speaker: Amaury Lambert, Laboratoire de Probabilités & Modèles Aléatoires, UPMC Univ Paris 06**

**Title: ** Random ultrametric trees and applications

**Abstract: ** Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and more recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.

**Week 2 - Monday 1st May**

**BANK HOLIDAY - No Seminar**

**Week 3 - Monday 8th May**

**Speaker: Jeffrey Rosenthal, University of Toronto**

**Title:** Conditions for Convergence of Adaptive MCMC Algorithms

**Abstract: ** Markov chain Monte Carlo (MCMC) algorithms, such as the Metropolis Algorithm and the Gibbs Sampler, are an extremely popular method of approximately sampling from complicated probability distributions. Adaptive MCMC attempts to automatically modify the algorithm while it runs, to improve its performance on the fly. However, such modifications can destroy the convergence properties necessary for the algorithm to be valid. In this talk, we first illustrate MCMC algorithms using simple graphical Java applets. We then discuss adaptive MCMC, and present examples and theorems concerning its convergence and efficiency. We close with some recent results which provide more easily verifiable sufficient conditions for convergence.

**Week 4 - Monday 15th May**

**Speaker:** **Matthias Winkel, Department of Statistics, University of Oxford**

**Title:** Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees

**Abstract:** In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have n leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have n vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergence to Gromov-Hausdorff-Prokhorov remains open for their models conditioned to have n vertices.

This is joint work with Hui He, Beijing Normal University.

**Week 5 - Monday 22nd May**

**Speaker:** **Mikolaj Kasprzak, Department of Statistics, University of Oxford
Title: ** Diffusion approximations via Stein's method and time changes

**We extend the ideas of (Barbour, 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. This allows us to bound the distance between the Moran model with mutation and Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-changed Poisson processes and the diffusive part of the latter may be expressed as a time-changed Brownian Motion. The method is applicable to a much wider class of examples satisfying the Stroock-Varadhan theory of diffusion approximation.**

**Abstract:**

**Week 6 - Monday 29th May**

**BANK HOLIDAY - No Seminar**

**Week 7 - Monday 5th June**

**No Seminar**

**Week 8 - Monday 12th June**

**No Seminar**

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**Week 9 - Monday 19th June (Lecture Room L6)**

** Speaker: ** Raphael Forien, CMAP - École Polytechnique, France

** Title: ** Gene flow across geographical barriers

** Abstract:** Barriers to gene flow are physical or biological obstacles which locally reduce migration and genetic exchanges between different regions of a species' habitat. Genetic relatedness between individuals can be deduced from the distribution of the positions of their ancestors back in time. In the presence of obstacles to migration, these ancestral lineages can be modelled by simple random walks outside of a bounded region around the obstacle. We present a continuous real-valued process which is obtained as a scaling limit of these random walks, and we give several constructions of this process.

Previous Workshops: HT17; MT16; TT16; HT16; MT15; TT15; HT15; MT14; TT14; HT14; MT13