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 t have a most recent common ancestor (MRCA) who lived at time A t , 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 N with time measured in N generations, i.e. in the scaling of population genetics which leads to Fisher–Wright diffusions and Kingman’s coalescent, we study the process A = ( A t ) whose jumps form the point process of time pairs ( 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 E as well as the times B 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.