## The Annals of Probability

### Results for the Stepping Stone Model for Migration in Population Genetics

Stanley Sawyer

#### Abstract

The stepping stone model describes a situation in which beasts alternately migrate among an infinite array of colonies, undergo random mating within each colony, and are subject to selectively neutral mutation at the rate $u$. Assume the beasts follow a random walk $\{X_n\}$. If $u = 0$, we show that two randomly chosen beasts in the $n$th generation in any bounded set are genetically identical at a given locus with probability converging to one iff the symmetrization of $\{X_n\}$ is recurrent. In general, if either $u = 0$ or $u$ is of order $1/n$, this probability converges to its limit at the rate $C/n^{\frac{1}{2}}$ for finite variance walks in one dimension and $C/(\log n)^a$ in two, with other rates for other classes of $\{X_n\}$. More complicated rates ensure for $u \neq O(1/n)$.

#### Article information

Source
Ann. Probab. Volume 4, Number 5 (1976), 699-728.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995980

Digital Object Identifier
doi:10.1214/aop/1176995980

Mathematical Reviews number (MathSciNet)
MR682605

Zentralblatt MATH identifier
0341.92009

JSTOR