J. F. C. Kingman

On the genealogy of large populations

A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain. WRIGHT-FISHER MODEL; NEUTRAL MUTATION; RANDOM EQUIVALENCE RELATIONS; COALESCENT; EWENS SAMPLING FORMULA; COUPLING; ENTRANCE BOUNDARY

Random partitions in population genetics

This paper is concerned with models for the genetic variation of a sample of gametes from a large population. The need for consistency between different sample sizes limits the mathematical possibilities to what are here called ‘partition structures Distinctive among them is the structure described by the Ewens sampling formula, which is shown to enjoy a characteristic property of non-interference between the different alleles. This characterization explains the robustness of the Ewens formula when neither selection nor recurrent mutation is significant, although different structures arise from selective and ‘charge-state’ models

The population structure associated with the Ewens sampling formula

Abstract Models for selectively neutral mutation, in which mutation always yields a new allele, seem always to lead, in the limit of large population size, to a sampling formula first propounded by Ewens in 1972. It is shown that the asymptotic validity of the Ewens formula is equivalent to a certain limiting joint distribution for the allele proportions in the population, arranged in descending order. The familiar diffusion approximations are corollaries of this limiting distribution, and therefore share the apparent robustness of the sampling formula.

Coherent random walks arising in some genetical models

Certain stochastic models used in population genetics have the form of Markov processes in which a group of N points moves randomly on a line, and in which an equilibrium distribution exists for the relative configuration of the group. The properties of this equilibrium are studied, with particular reference to a certain limiting situation as N becomes large. In this limit the group of points is distributed like a large sample from a distribution which is itself subject to random variation.

Markov transition probabilities. I

7. ConclusionThe most important results of this paper can now be summed up in the following assertion.There is a convex cone ℳ of measures on (0, ∞), closed under countable sums and finite convolutions (when defined), and containing only measures equivalent to Lebesgue measure, such that(i)a function p is a diagonal Markov function if and only if it is continuous and satisfies for all positive θ the equation

Introduction to Measure and Probability.

\int\limits_0^\infty {p(t)e^{ - \theta t} dt} = \{ \theta + \int {(1 - e^{\theta x} )\mu (dx)\} ^{ - 1} ,}

Stationary regenerative phenomena

where Μ is a measure on (0, ∞] whose inner part belongs to ℳ;(ii)a function p is a non-diagonal Markov function if and only if it can be written if the form where and are diagonal Markov functions, λ is a totally finite measure on [0, ∞) whose inner part belongs to ℳ, and