#### Date of Award

Spring 2012

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Supervisor

Shui Feng

#### Language

English

#### Committee Member

Fred M. Hoppe, Roman Viveros-Aguilera

#### Abstract

In this thesis, we explore several stochastic models associated withthe two-parameter Poisson-Dirichlet distribution and population genetics.The impacts of mutation, selection and time onthe population evolutionary process will be studied by focusing on two aspects of the model:equilibrium and non-equilibrium. In the first chapter, we introduce relevant background on stochastic genetic models, andsummarize our main results and their motivations. In the second chapter, the two-parameter GEM distribution is constructedfrom a linear birth process with immigration. The derivationrelies on the limiting behavior of the age-ordered family frequencies. In the third chapter, to show the robustness of the sampling formula we derive the Laplace transform of the two-parameterPoisson-Dirichlet distribution from Pitman sampling formula. The correlationmeasure of the two-parameter point process is obtained in our proof. We also reverse this derivationby getting the sampling formula from the Laplace transform. Then,we establish a central limit theorem for the infinitely-many-neutral-alleles modelat a fixed time as the mutation rate goes to infinity.Lastly, we get the Laplace transform for the selectionmodel from its sampling formula. In the fourth chapter, we establisha central limit theorem for the homozygosity functions under overdominant selectionwith mutation approaching infinity. The selection intensity is given by a multiple of certain powerof the mutation rate. This result shows an asymptotic normality for the properly scaled homozygosities,resembling the neutral model without selection.This implies that the influence of selection can hardly be observed with large mutation. In the fifth chapter, the stochastic dynamics of the two-parameter extension of theinfinitely-many-neutral-alleles model is characterized by the derivation of its transition function,which is absolutely continuous with respect to the stationary distribution being the two-parameter Poisson-Dirichlet distribution.The transition density is obtained by the expansion of eigenfunctions.Combining this result with the correlation measure in Chapter 3, we obtain the probability generatingfunction of a random sampling from the two-parameter model at a fixed time. Finally, we obtain two results based on the quasi-invariance of the Gamma processwith respect to the multiplication transformation group.One is the quasi-invariance property of the two-parameter Poisson-Dirichletdistribution with respect to Markovian transformation group.The other one is the equivalence between the quasi-invarianceof the stationary distributions of aclass of branching processes and their reversibility.

#### Recommended Citation

Xu, Fang, "STOCHASTIC MODELS ASSOCIATED WITH THE TWO-PARAMETER POISSON-DIRICHLET DISTRIBUTION" (2012). *Open Access Dissertations and Theses.* Paper 6797.

http://digitalcommons.mcmaster.ca/opendissertations/6797

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