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Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is(More)
(2000). A transition function expansion for a diffusion model with selection. Abstract Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of(More)
The Wright--Fisher model and its approximating diffusion model are compared in terms of the expected value of a smooth but arbitrary function of nth-generation gene frequency. In the absence of selection, this expectation is shown to differ in the two models by at most a linear combination (with coefficients depending only on the derivatives of the smooth(More)
Fleming᎐Viot processes are probability-measure-valued diffusion processes that can be used as stochastic models in population genetics. Here we use duality methods to prove ergodic theorems for Fleming᎐Viot processes, including those with recombination. Coupling methods are also used to establish ergodicity of Fleming᎐Viot processes, first without and then(More)
Petrov constructed a diffusion process in the compact Kingman simplex whose unique stationary distribution is the two-parameter Poisson–Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates sum to 1 is the natural state space for the process. In fact, the complementary set acts like an entrance(More)
There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have developed(More)
That there exist two losing games that can be combined, either by random mixture or by nonran-dom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a(More)
This study considers configurations of identity-by-descent sharing in sibships as an alternative to breaking the sibship into sib pairs. A general method is presented for specifying all distinct configurations and deriving their probabilities. The method works for sibships of arbitrary size and can be implemented with paper and pencil. Specific results are(More)