Solving the fisher-wright and coalescence problems with a discrete markov chain analysis

We develop a new, self-contained proof that the expected number of generations required for gene allele fixation or extinction in a population of size n is O(n) under general asumptions. 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 establish On H (p)) as the expected time for gene allele fixation or extinction for the Fisher-Wright problem, where the gene occurs with initial frequency p and H (p) is the entropy function. Under a weaker hypothesis on the variance, the expected time is O(n(P(1 - p)) /3) for fixation or extinction. Thus, the expected-time bound of 0 (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 size n with n distinct alleles is shown to be O (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.