变异Cannings模型平衡分布的Dirichlet逼近
Dirichlet approximation of equilibrium distributions in Cannings models with mutation
摘要：
考虑一个具有有限数量等位基因类型的固定有限大小的单倍体群体，该群体具有中性突变的Cannings可交换系谱。各世代等位基因数的马尔可夫链的平稳分布是群体遗传学中的一个重要量，但一般没有易处理的描述。对于具有一般突变结构的Wright-Fisher系谱和具有双亲独立突变结构的Cannings可交换系谱，我们给出了Dirichlet分布与这个有限总体平稳分布之间的分布距离的上界.在第一种情况下，如果群体很大并且突变不太依赖于亲本类型，则界限很小;“太多”自然由我们的界限量化。在第二种情况下，如果人口很大，并且Cannings谱系中三次合并的机会相对于两次合并的机会很小，则界限很小;这是确保谱系收敛到Kingman合并的相同条件。这些结果遵循一个新的发展斯坦的方法的狄利克雷分布的基础上Barthel的发电机方法和概率描述的半群的赖特-费舍尔扩散由于格里菲斯，李和Tavaré。
英文摘要：
Consider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite population stationary distribution for the Wright-Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent mutation structure. In the first case, the bound is small if the population is large and the mutations do not depend too much on parent type; "too much" is naturally quantified by our bound. In the second case, the bound is small if the population is large and the chance of three-mergers in the Cannings genealogy is small relative to the chance of two-mergers; this is the same condition to ensure convergence of the genealogy to Kingman's coalescent. These results follow from a new development of Stein's method for the Dirichlet distribution based on Barbour's generator approach and a probabilistic description of the semigroup of the Wright-Fisher diffusion due to Griffiths, and Li and Tavaré.
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