Cannings models, population size changes and multiple-merger coalescents

Multiple-merger coalescents, e.g. [Formula: see text]-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. [Formula: see text]-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 [Formula: see text]. 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 [Formula: see text]-n-coalescents, this has been explicitly shown for only a limited subclass of [Formula: see text]-n-coalescents and exponentially growing populations. This article gives a more general construction of time-changed [Formula: see text]-n-coalescents as limits of specific Cannings models with rather arbitrary time changes.