Cannings models, population size changes and multiple-merger coalescents

Abstract

Multiple-merger coalescents, e.g. $\Lambda$-$n$-coalescents, have been proposed as models of the genealogy of $n$ sampled individuals for a range of populations whose genealogical structures are not captured well by Kingman's $n$-coalescent. $\Lambda$-$n$-coalescents can be seen as the limit process of the discrete genealogies of Cannings models with fixed population size, when time is rescaled and population size $N\to \infty$. As established for Kingman's $n$-coalescent, moderate population size fluctuations in the discrete population model should be reflected by a time-change of the limit coalescent. For $\Lambda$-$n$-coalescents, this has been explicitly shown for only a limited subclass of $\Lambda$-$n$-coalescents and exponentially growing populations. This article gives a general construction of time-changed $\Lambda$-$n$-coalescents as limits of specific Cannings models with rather arbitrary time changes.