## The Annals of Probability

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

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

#### 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: 19 April 2007

**Permanent link to this document**

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

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176995980

**Mathematical Reviews number (MathSciNet)**

MR682605

**Zentralblatt MATH identifier**

0341.92009

**Subjects**

Primary: 92A10

Secondary: 92A15 60J15 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60K99: None of the above, but in this section

**Keywords**

Stepping stone model random walks genetics population genetics diploid migration mutation random mating rate of convergence

#### Citation

Sawyer, Stanley. Results for the Stepping Stone Model for Migration in Population Genetics. The Annals of Probability 4 (1976), no. 5, 699--728. doi:10.1214/aop/1176995980. http://projecteuclid.org/euclid.aop/1176995980.