%0 Journal Article %1 sawyer1976 %A Sawyer, Stanley %D 1976 %I The Institute of Mathematical Statistics %J The Annals of Probability %K Karamatas_theorem history isolation_by_distance long_distance_dispersal range_expansions spatial_coalescent stable_distributions %N 5 %P 699--728 %R 10.1214/aop/1176995980 %T Results for the Stepping Stone Model for Migration in Population Genetics %U http://dx.doi.org/10.1214/aop/1176995980 %V 4 %X The stepping stone model descrbies 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 $C/n$ for finite variance walks in one dimension and $C/(n)^a$ in two, with other rates for other classes of $\X_n\$. More complicated rates ensue for $u O(1/n)$.

@article{sawyer1976, abstract = {The stepping stone model descrbies 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 $C/\sqrt{n}$ 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 ensue for $u \neq O(1/n)$.}, added-at = {2014-07-23T22:31:55.000+0200}, ajournal = {Ann. Probab.}, author = {Sawyer, Stanley}, biburl = {https://www.bibsonomy.org/bibtex/20482e628236a06d63c6aee6893c22fd5/peter.ralph}, doi = {10.1214/aop/1176995980}, interhash = {f2258ff7f437362cf0a95c4582bd7422}, intrahash = {0482e628236a06d63c6aee6893c22fd5}, journal = {The Annals of Probability}, keywords = {Karamatas_theorem history isolation_by_distance long_distance_dispersal range_expansions spatial_coalescent stable_distributions}, month = {10}, number = 5, pages = {699--728}, publisher = {The Institute of Mathematical Statistics}, timestamp = {2015-07-31T00:52:57.000+0200}, title = {Results for the Stepping Stone Model for Migration in Population Genetics}, url = {http://dx.doi.org/10.1214/aop/1176995980}, volume = 4, year = 1976 }