We derive conditions under which a sequence of processes will converge to a (continuous-time) Markov chain with an entrance boundary. Our main application of this result is in proving weak convergence of the so-called population ancestral processes, associated with a wide class of exchangeable reproductive models, to a particular death process with an entrance boundary at infinity. This settles a conjecture of Kingman. We also prove weak convergence of the absorption times of many neutral genetics models to that of the Wright-Fisher diffusion, and convergence of population line-of-descent processes to another death process.
Peter Donnelly. "Weak Convergence to a Markov Chain with an Entrance Boundary: Ancestral Processes in Population Genetics." Ann. Probab. 19 (3) 1102 - 1117, July, 1991. https://doi.org/10.1214/aop/1176990336