We consider a sequence of Markov chains occurring in population genetics (viz., the so-called multiallelic Wright-Fisher models) that converges weakly to a multidimensional diffusion process. Certain absorption times, which arise naturally in connection with the genetic models, are shown to also converge weakly. This extends a result of Guess. Corollaries include convergence of moments of absorption times and convergence of absorption probabilities. The latter results are used implicitly in population genetics.
"Limit theorems for Absorption Times of Genetic Models." Ann. Probab. 7 (4) 622 - 638, August, 1979. https://doi.org/10.1214/aop/1176994986