The Poisson-Dirichlet Distribution and Related Topics (eBook)

Shui Feng is a Professor of Mathematics and Statistics at McMaster University, Canada. His research is in the broad area of stochastic processes and limit theorems, with particular interests in stochastic models in population genetics, finance, and statistical physics. He is the co-editor of one book and the author of more than 30 research papers.

Preface 7
Contents 11
Part I Models 14
Introduction 15
Discrete Models 15
Genetic Background 15
The Wright--Fisher Model 17
The Moran Model 19
Diffusion Approximation 20
An Important Relation 23
Notes 25
The Poisson--Dirichlet Distribution 27
Definition and Poisson Process Representation 27
Perman's Formula 29
Marginal Distribution 34
Size-biased Sampling and the GEM Representation 36
The Ewens Sampling Formula 38
Scale-invariant Poisson Process 45
Urn-based Models 48
Hoppe's Urn 48
Linear Birth Process with Immigration 50
A Model of Joyce and Tavaré 56
The Dirichlet Process 58
Notes 63
The Two-Parameter Poisson--Dirichlet Distribution 65
Definition 65
Marginal Distributions 66
The Pitman Sampling Formula 70
Urn-type Construction 74
Notes 78
The Coalescent 79
Kingman's n-Coalescent 79
The Coalescent 84
The Pure-death Markov Chain 85
Notes 92
Stochastic Dynamics 93
Infinitely-many-neutral-alleles Model 93
A Fleming--Viot Process 97
The Structure of Transition Functions 105
A Measure-valued Branching Diffusion with Immigration 119
Two-parameter Generalizations 121
Notes 123
Particle Representation 125
Exchangeability and Random Probability Measures 125
The Moran Process and the Fleming--Viot Process 128
The Donnelly--Kurtz Look-down Process 132
Embedded Coalescent 135
Notes 136
Part II Asymptotic Behaviors 137
Fluctuation Theorems 138
The Poisson--Dirichlet Distribution 138
The Dirichlet Process 145
Gaussian Limits 153
Notes 161
Large Deviations for the Poisson--Dirichlet Distribution 162
Large Mutation Rate 162
The Poisson--Dirichlet Distribution 162
The Two-parameter Poisson--Dirichlet Distribution 169
Small Mutation Rate 171
The Poisson--Dirichlet Distribution 171
Two-parameter Generalization 179
Applications 182
Notes 189
Large Deviations for the Dirichlet Processes 190
One-parameter Case 190
Two-parameter Case 198
Comparison of Rate Functions 204
Notes 208
Poisson Process and Poisson Random Measure 209
Definitions 209
Properties 210
Basics of Large Deviations 213
References 219
Index 227

"Chapter 1 Introduction (p. 3-4)

In this chapter, we introduce several basic models in population genetics including the Wright–Fisher model, the Moran model, and the corresponding diffusion approximations. The Dirichlet distribution is introduced as the reversible measure of the corresponding diffusion processes. Its connection to the gamma distribution is explored. These will provide the necessary intuition and motivation for the Poisson– Dirichlet distribution and other sophisticated models considered in subsequent chapters.

1.1 Discrete Models

We begin this section with a brief introduction of the genetic terminology used throughout the book.

1.1.1 Genetic Background

All living organisms inherit biological characteristics from their parents. The characteristics that can be inherited are called the genetic information. Genetics is concerned with the study of heredity and variation. Inside each cell of an organism, there is a fixed number of chromosomes, which are threadlike objects, each containing a single, long molecule called DNA (deoxyribonucleic acid). Each DNA molecule is composed of two strands of nucleotides in which the sugar is deoxyribose and the bases are adenine (A), cytosine (C), guanine (G), and thymine (T).

Linked by hydrogen bonds with A paired with T, and G paired with C, the two strands are coiled around to form the famous double helix structure discovered by Watson and Crick in 1953. These DNA molecules are responsible for the storage and inheritance of genetic information. A gene is a hereditary unit composed of a portion of DNA. The place that a gene resides on a chromosome is called a locus (loci in plural form). Different forms of a gene are called alleles.

An example is the ABO blood group locus, admitting three alleles, A, B, and O. The complete set of genetic information of an organism is called a genome. An organism is called haploid if there is only one set of chromosomes; if chromosomes appear in pairs, the organism is diploid. The set of chromosomes in a polyploid organism is at least tripled. Bacteria and fungi are generally haploid organisms, whereas most higher organisms such as mammals are diploid.

Human diploid cells have 46 (or 23 pairs) chromosomes and human haploid gametes (egg and sperm) each have 23 chromosomes. Bread wheat and canola have six and four sets of chromosomes, respectively. The genotype of an individual at a particular locus is the set of allele(s) presented. For a diploid individual, this is an unordered pair of alleles (one on each chromosome). A diploid individual is homozygous (heterozygous) at a particular locus if the corresponding genotype consists of two identical alleles (different alleles). A phenotype of an organism is any observable characteristic. Phenotypes are determined by an organism’s genotypes and the interaction of the genotypes with the environment."