The process of most recent common ancestors in an evolving coalescent

Consider a haploid population which has evolved through an exchangeable reproduction dynamics, and in which all individuals alive at time tt have a most recent common ancestor (MRCA) who lived at time AtAt, say. As time goes on, not only the population but also its genealogy evolves: some families will get lost from the population and eventually a new MRCA will be established. For a time stationary situation and in the limit of infinite population size NN with time measured in NN generations, i.e. in the scaling of population genetics which leads to Fisher–Wright diffusions and Kingman’s coalescent, we study the process A=(At)A=(At) whose jumps form the point process of time pairs (E,B)(E,B) when new MRCAs are established and when they lived. By representing these pairs as the entrance and exit time of particles whose trajectories are embedded in the look-down graph of Donnelly and Kurtz [P. Donnelly, T.G. Kurtz, Particle representations for measure-valued population models, Ann. Probab. 27 (1) (1999) 166–205] we can show by exchangeability arguments that the times EE as well as the times BB form a Poisson process. Furthermore, the particle representation helps to compute various features of the MRCA process, such as the distribution of the coalescent at the instant when a new MRCA is established, and the distribution of the number of MRCAs to come that live in today’s past.