Solving the Fisher-Wright and coalescence problems with a discrete Markov chain analysis

Abstract

We develop a new, self-contained proof that the expected number of generations required for gene allele fixation or extinction in a population of sizenisO(n) under general assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixation or extinction time to any desired precision. Our proofs establishO(nH(p)) as the expected time for gene allele fixation or extinction for the Fisher-Wright problem, where the gene occurs with initial frequencypandH(p) is the entropy function. Under a weaker hypothesis on the variance, the expected time isO(n(p(1-p))1/2) for fixation or extinction. Thus, the expected-time bound ofO(n) for fixation or extinction holds in a wide range of situations. In the multi-allele case, the expected time for allele fixation or extinction in a population of sizenwithndistinct alleles is shown to beO(n). From this, a new proof is given of a coalescence theorem about the mean time to the most recent common ancestor (MRCA), which applies to a broad range of reproduction models satisfying our mean and weak variation conditions.