临界与下临界多物种分支过程的家谱树极限

发布时间：2018-07-02 12:09

  本文选题：下临界的多物种分支过程 + 临界的多物种分支过程　； 参考：《南京大学》2017年硕士论文

【摘要】：分支过程从由一百多年前因研究姓氏消亡问题而产生,到如今在生物学与物理学中的运用广泛,可谓是成就斐然。过程分类也由最开始的单物种离散时间的基本情形,逐渐开拓到多物种、连续时间、带有移民等复杂情况。本篇论文主要通过对下临界、临界、上临界三种不同的多物种离散时间的Galton-WWatson分支过程的性质对比研究来讨论他们从久远的后代回溯到遥远的祖先的物种种类的极限情况,最终目的是寻求每种过程家谱树的Markov极限。本文中有关临界与下临界情况的讨论是在第n代时此过程依旧没有灭绝的前提假设下展开的,因为这两种过程的灭绝概率为1,最终必然会灭绝。而上临界过程由于灭绝概率小于1,所以不做此假设。论文最终使用数学归纳法推导得到了这三种过程分别对应的家谱系的Markov极限,其中主要利用了上临界过程的鞅收敛性质,临界过程时退化的物种比例性质和下临界过程的特殊的极限分布测度的性质。最后,文章给出了此极限Markov链的初始分布、转移概率和平稳分布。
[Abstract]:The branching process began more than 100 years ago because of the death of surnames, and now it has been widely used in biology and physics, and it can be described as a great achievement. The process classification is also developed from the basic case of discrete time of single species at the beginning to complex situations such as multi-species, continuous time and immigration. In this paper, we discuss the limit of species from their distant progeny to distant ancestors by comparing the properties of three different multispecies discrete time Galton-WWatson branching processes. The ultimate goal is to find the Markov limit of each process tree. The discussion of the critical and lower critical conditions in this paper is carried out under the assumption that there is still no extinction in the n-generation process, because the extinction probability of these two processes is 1, and it is inevitable that the two processes will eventually become extinct. The upper critical process does not make this assumption because the extinction probability is less than 1. Finally, the Markov limit of the family pedigree corresponding to these three processes is derived by mathematical induction, in which the martingale convergence property of the supercritical process is mainly used. The properties of the degenerate species proportional property and the special limit distribution measure of the lower critical process during the critical process. Finally, the initial distribution, transition probability and stationary distribution of the limit Markov chain are given.
【学位授予单位】：南京大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.65
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本文编号：2090074