Solving the Fisher-wright and Coalescence Prob- Lems 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 assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixation/extinction time to any desired precision. Our proofs establish O(nH(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)) for fixation or extinction. Thus, the expected time bound of O(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 mean time to most recent common ancestor (MRCA) that applies to a broad range of reproduction models satisfying our mean and weak variation conditions.