On the spectrum and ergodicity of a neutral multi-allelic Moran model

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
J. Corujo
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引用次数: 2

Abstract

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.
中性多等位基因Moran模型的谱和遍历性
本文的目的是给出中性多型Moran模型的特征值的完整描述,以及在平稳收敛速度研究中的应用。我们考虑的Moran模型是一个具有未知平稳分布的一般不可逆连续马尔可夫链。具体地说,我们考虑$N$个个体,其中每个个体都属于$K$个可能的等位基因类型中的一种类型。个体以两种方式相互作用:一种是独立的不可还原的突变过程,另一种是繁殖过程,即随机选择一对个体,其中一个死亡,另一个繁殖。我们的主要结果提供了Moran过程的无限小发生器矩阵的特征值的显式表达式,用跳跃率矩阵的特征值表示。作为这一结果的结果,我们研究了过程的总变化收敛到平稳,并给出了Moran过程混合时间的下界。此外,我们还详细研究了具有亲本独立突变方案的中性多等位基因Moran模型的谱分解,该突变方案是使中性Moran过程可逆的唯一突变方案。在亲本独立突变条件下,我们还证明了在初始所有个体均为同一类型且个体数趋于无穷大时,卡方和总变异距离存在截断现象。此外,在没有繁殖的情况下,我们证明了初始所有个体都是同一类型时,亲本独立突变过程的总变异距离具有高斯分布。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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