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 $\mathcal{A}=(A_t)$ 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 (1999) we can show by exchangeability arguments that the times $E$ as well as the times $B$ from 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.