arXiv subjects

Shui Feng

Publications and source records attributed to Shui Feng.

25 records · Page 2Linked to original sources

arXiv2008

Poisson-Dirichlet Distribution with Small Mutation Rate

The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero.

math.PR↗

arXiv2008

Reversibility of Interacting Fleming-Viot Processes with Mutation, Selection, and Recombination

Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

math.PR↗

arXiv2007

A Class of Infinite Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1\}$ with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space $S$. This provides a reasonable alternative to the Fleming-Viot process which does not satisfy the log-Sobolev inequality when $S$ is infinite as observed by W. Stannat \cite{S}.

math.PR↗

arXiv2007

Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula

Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $θ$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $θ$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $θ$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson--Dirichlet distribution in terms of large deviations. Large $θ$ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.

math.PR↗

arXiv2006

Large deviations for Dirichlet processes and Poisson-Dirichlet distributions with two parameters

Large deviation principles are established for the two-parameter Poisson-Dirichlet distribution and two-parameter Dirichlet process when parameter $θ$ approaches infinity. The motivation for these results is to understand the differences in terms of large deviations between the two-parameter models and their one-parameter counterparts. New insight is obtained about the role of the second parameter $α$ through a comparison with the corresponding results for the one-parameter Poisson-Dirichlet distribution and Dirichlet process.

math.PR↗

arXiv2006

Asymptotic behavior of the Poisson--Dirichlet distribution for large mutation rate

The large deviation principle is established for the Poisson--Dirichlet distribution when the parameter $θ$ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson--Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in $θ$, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in $θ$, then the large deviation rate function includes an additional term coming from selection. The application of these results to the heterozygote advantage model provides an alternate proof of one of Gillespie's conjectures in [Theoret. Popul. Biol. 55 145--156].

math.PR↗

arXiv1998

Large deviations for the Fleming-Viot process with neutral mutation and selection

Large deviation principles are established for the Fleming-Viot processes with neutral mutation and selection, and the corresponding equilibrium measures as the sampling rate goes to 0. All results are first proved for the finite allele model, and then generalized, through the projective limit technique, to the infinite allele model. Explicit expressions are obtained for the rate functions.

math.PR↗