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October, 1976 Results for the Stepping Stone Model for Migration in Population Genetics
Stanley Sawyer
Ann. Probab. 4(5): 699-728 (October, 1976). DOI: 10.1214/aop/1176995980


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 $u \neq O(1/n)$.


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Stanley Sawyer. "Results for the Stepping Stone Model for Migration in Population Genetics." Ann. Probab. 4 (5) 699 - 728, October, 1976.


Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0341.92009
MathSciNet: MR682605
Digital Object Identifier: 10.1214/aop/1176995980

Primary: 92A10
Secondary: 60J15 , 60J20 , 60K99 , 92A15

Keywords: diploid , Genetics , migration , mutation , Population genetics , random mating , Random walks , rate of convergence , Stepping stone model

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 5 • October, 1976
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