%0 Journal Article
%T Fluctuations Analysis of Finite Discrete Birth and Death Chains with Emphasis on Moran Models with Mutations
%A Thierry E. Huillet
%J ISRN Biomathematics
%D 2013
%R 10.1155/2013/939308
%X The Moran model is a discrete-time birth and death Markov chain describing the evolution of the number of type 1 alleles in a haploid population with two alleles whose total size is preserved during the course of evolution. Bias mechanisms such as mutations or selection can affect its neutral dynamics. For the ergodic Moran model with mutations, we get interested in the fixation probabilities of a mutant, the growth rate of fluctuations, the first hitting time of the equilibrium state starting from state , the first return time to the equilibrium state, and the first hitting time of starting from , together with the time needed for the walker to reach its invariant measure, again starting from . For the last point, an appeal to the notion of Siegmund duality is necessary, and a cutoff phenomenon will be made explicit. We are interested in these problems in the large population size limit . The Moran model with mutations includes the heat exchange models of Ehrenfest and Bernoulli-Laplace as particular cases; these were studied from the point of view of the controversy concerning irreversibility ( -theorem) and the recurrence of states. 1. Introduction In Section 2, we start with generalities on discrete-time birth and death Markov chains with finite state space, whose transition matrix is of Jacobi type: spectral representation, reversibility versus -theorem, large deviations of the empirical average, invariant measure, conditions of transience/recurrence, spectral positivity and stochastic monotonicity, distribution of hitting and first return times, and Greens function. The scale or harmonic function is shown to be of interest to determine the excursions heights would the chain be ergodic. After having fixed the background, we then proceed with the study of specific Moran chains presenting various bias mechanisms. The Moran chain model is a particular instance of a discrete-time birth and death Markov chain, describing the evolution of the number of type alleles in an haploid population with two alleles whose total size is preserved during the course of evolution. Bias mechanisms such as mutations or selection can affect its neutral dynamics (there are alternative domains of science where this model makes sense, namely, economy and behavioral sciences. Here, the Markov state takes on the interpretation of the number of agents that belong to the first strategy (say right- or left-wing voters) at time . See [1, 2] and references therein for a recent study with this physical image in mind.). For the ergodic Moran model with mutations [3], we get
%U http://www.hindawi.com/journals/isrn.biomathematics/2013/939308/