Journal article:

Samuel R. Buss and Peter Clote
"Solving the Fisher-Wright and coalescence problems with a discrete Markov chain analysis"
Advances in Probability Theory 36, 4 (2004) 1175–1197.

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We develop a new, self-contained proof, based on a discrete Markov chain analysis, that the expected number of generations required for gene allele fixation or extinction in a population of size n is O(n) under the assumption of neutral selection. Such problems, collectively known as the Fisher-Wright problem, had previously been shown to have expected linear time solutions; previous proofs were technically quite difficult and based on an approximation of the discrete Fisher-Wright problem by a continuous model involving the diffusion equation (Fisher~\cite{fisher:1930}, Wright~\cite{wright:1945,wright:1949}, Kimura~\cite{kimura:1955,kimura:1962,kimura:1964},  Watterson~\cite{watterson:1962}, Ewens~\cite{ewens:1963}). In contrast, our new proof is direct, self-contained, and relies on a discrete Markov chain analysis. We further develop an algorithm to compute the 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 frequency p and H(p) is the entropy function. We introduce a weaker hypothesis on the standard deviation and prove an expected time of O(n\cdot \sqrt{p(1-p)}) for fixation or extinction under this weaker hypothesis. Thus, the expected time bound of O(n) for fixation or extinction holds in a wider range of situations than have been considered previously.
Additionally, we study the coalescence problem and prove that the expected time for allele fixation or extinction in a population of size n with  n distinct alleles is  O(n). Similar bounds are well-known from coalescent theory for the Fisher-Wright model; however, our results apply to a broader range of reproduction models that satisfy our mean condition and weak variation condition.

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