On the genealogy of multi-type Cannings models and their limiting exchangeable coalescents

We study the multi-type Cannings population model. Each individual has a type belonging to a given at most countable type space $E$. The population is hence divided into $\mathrm{\mid }E\mathrm{\mid }$ subpopulations. The subpopulation sizes are assumed to be constant over the generations, whereas the number of offspring of type $\mathrm{\ell }\in E$ of all individuals of type $k\in E$ is allowed to be random. Under a joint exchangeability assumption on the offspring numbers, the transition probabilities of the ancestral process of a sample of individuals satisfy a multi-type consistency property, paving a way to prove in the limit for large subpopulation sizes the existence of multi-type exchangeable coalescent processes via Kolmogorov’s extension theorem. Integral representations for the infinitesimal rates of these multi-type exchangeable coalescents and some of their properties are studied. Examples are provided, among them multi-type Wright-Fisher models and multi-type pure mutation models. The results contribute to the foundations of multi-type coalescent theory and provide new insights into (the existence of) multi-type exchangeable coalescents.