Next Article in Journal
Generalized Hyers-Ulam Stability of Trigonometric Functional Equations
Next Article in Special Issue
Analysis of the Incidence of Poxvirus on the Dynamics between Red and Grey Squirrels
Previous Article in Journal
A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation
Previous Article in Special Issue
The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

## Printed Edition

A printed edition of this Special Issue is available at MDPI Books....
Open AccessArticle

# Linearization of the Kingman Coalescent

Computational Biology and Bioinformatics Unit, Research School of Biology, R.N. Robertson Building 46, Australian National University, Canberra, ACT 0200, Australia
Mathematics 2018, 6(5), 82; https://doi.org/10.3390/math6050082
Received: 5 February 2018 / Revised: 19 April 2018 / Accepted: 9 May 2018 / Published: 14 May 2018
Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term $( n 2 )$ where n denotes the number of lineages present in the genealogy. These two standard constraints of Kingman’s coalescent, obtained in the limit of a large population size, approximate the exact ancestral process of Wright-Fisher or Moran models under appropriate parameterization. Calculation of coalescence event probabilities with higher accuracy quantifies the dependence of sample and population sizes that adhere to Kingman’s coalescent process. The convention that probabilities of leading order $N − 2$ are negligible provided $n ≪ N$ is examined at key stages of the mathematical derivation. Empirically, expected genealogical parity of the single-pair restricted Wright-Fisher haploid model exceeds 99% where $n ≤ 1 2 N 3$ ; similarly, per expected interval where $n ≤ 1 2 N / 6$ . The fractional cubic root criterion is practicable, since although it corresponds to perfect parity and to an extent confounds identifiability it also accords with manageable conditional probabilities of multi-coalescence. View Full-Text
Show Figures

Figure 1

MDPI and ACS Style

Slade, P.F. Linearization of the Kingman Coalescent. Mathematics 2018, 6, 82.