本文選題：下臨界的多物種分支過程 + 臨界的多物種分支過程��； 參考：《南京大學(xué)》2017年碩士論文

【摘要】：分支過程從由一百多年前因研究姓氏消亡問題而產(chǎn)生,到如今在生物學(xué)與物理學(xué)中的運(yùn)用廣泛,可謂是成就斐然。過程分類也由最開始的單物種離散時間的基本情形,逐漸開拓到多物種、連續(xù)時間、帶有移民等復(fù)雜情況。本篇論文主要通過對下臨界、臨界、上臨界三種不同的多物種離散時間的Galton-WWatson分支過程的性質(zhì)對比研究來討論他們從久遠(yuǎn)的后代回溯到遙遠(yuǎn)的祖先的物種種類的極限情況,最終目的是尋求每種過程家譜樹的Markov極限。本文中有關(guān)臨界與下臨界情況的討論是在第n代時此過程依舊沒有滅絕的前提假設(shè)下展開的,因為這兩種過程的滅絕概率為1,最終必然會滅絕。而上臨界過程由于滅絕概率小于1,所以不做此假設(shè)。論文最終使用數(shù)學(xué)歸納法推導(dǎo)得到了這三種過程分別對應(yīng)的家譜系的Markov極限,其中主要利用了上臨界過程的鞅收斂性質(zhì),臨界過程時退化的物種比例性質(zhì)和下臨界過程的特殊的極限分布測度的性質(zhì)。最后,文章給出了此極限Markov鏈的初始分布、轉(zhuǎn)移概率和平穩(wěn)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.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O211.65
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本文編號：2090074