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.
@article{1176990336,
author = {Donnelly, Peter},
title = {Weak Convergence to a Markov Chain with an Entrance Boundary: Ancestral Processes in Population Genetics},
journal = {Ann. Probab.},
volume = {19},
number = {4},
year = {1991},
pages = { 1102-1117},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990336}
}
Donnelly, Peter. Weak Convergence to a Markov Chain with an Entrance Boundary: Ancestral Processes in Population Genetics. Ann. Probab., Tome 19 (1991) no. 4, pp. 1102-1117. http://gdmltest.u-ga.fr/item/1176990336/