Open Access
February, 1992 On the Stationary Distribution of the Neutral Diffusion Model in Population Genetics
S. N. Ethier, Thomas G. Kurtz
Ann. Appl. Probab. 2(1): 24-35 (February, 1992). DOI: 10.1214/aoap/1177005769

Abstract

Let S be a compact metric space, let θ>0, and let P(x,dy) be a one-step Feller transition function on S×B(S) corresponding to a weakly ergodic Markov chain in S with unique stationary distribution ν0. The neutral diffusion model, or Fleming-Viot process, with type space S, mutation intensity 12θ and mutation transition function P(x,dy), assumes values in P(S), the set of Borel probability measures on S with the topology of weak convergence, and is known to be weakly ergodic and have a unique stationary distribution ΠP(P(S)). Define the Markov chain {X(τ),τZ+} in S2S3 as follows. Let X(0)=(ξ,ξ)S2, where ξ is an S-valued random variable with distribution ν0. From state (x1,,xn)Sn, where n2, one of two types of transitions occurs. With probability θ/(n(n1+θ)) a transition to state (x1,,xi1,ξi,xi+1,,xn)Sn occurs (1in), where ξi is distributed according to P(xi,dy). With probability (n1)/((n+1)n(n1+θ)) a transition to state (x1,,xj1,xi,xj,,xn)Sn+1 occurs (1in,1jn+1). Letting τn denote the hitting time of Sn, we show that the empirical measure determined by the n coordinates of X(τn+11) converges almost surely as n to a P(S)-valued random variable with distribution Π.

Citation

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S. N. Ethier. Thomas G. Kurtz. "On the Stationary Distribution of the Neutral Diffusion Model in Population Genetics." Ann. Appl. Probab. 2 (1) 24 - 35, February, 1992. https://doi.org/10.1214/aoap/1177005769

Information

Published: February, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0742.60076
MathSciNet: MR1143391
Digital Object Identifier: 10.1214/aoap/1177005769

Subjects:
Primary: 60G57
Secondary: 60J70 , 92A10

Keywords: Fleming-Viot process , Genealogical tree , infinitely-many-sites model , measure-valued diffusion , Population genetics

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 1 • February, 1992
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