臺灣博碩士論文加值系統

這篇論文著重於於非齊次多型態分支過程，並且特別強調週期為二的情況。
在第一章中，我們簡單介紹了分支過程的歷史，並給出一些與Galton-Watson 分支過程和多型分支過程相關的符號和重要定理。
在第二章中，我們研究了週期為二的多型分支過程，發現了在偶數代和奇數代的長期情況下，族群的數量會成指數增長。具體來說，我們還發現，增長率和型態分解與某些關鍵矩陣的特徵值和特徵向量有關。
在第三章中，我們討論了族群的祖先型態的分佈。通過從族群中選取一個個體並向前追溯其祖先的型態，我們發現祖先型態分佈的極限呈現出某些馬爾可夫性質。

This thesis focuses on non-homogeneous multi-type branching processes, with a particular emphasis on those with period of two.
In Chapter 1, we provide a brief introduction to the history of branching processes, along with some notation and important theorems related to Galton-Watson branching processes and multitype branching processes.
In Chapter 2, we deal with 2-periodic multitype branching processes and discover that the exponential growth rates and the type decompositions of the population in the long run of even-numbered generations and odd-numbered generations. Precisely, we also find that the growth rates and type decompositions are related to the eigenvalues and eigenvectors of some key matrix.
In Chapter 3, we discuss the ancestral type distribution of the population. By selecting an individual from the population and tracing the types of its ancestors backward in time, we find that the limit of the distribution on the ancestral types exhibits some Markov properties.

中文摘要 i
Abstract ii
Contents iii

1 Introduction 1
1.1 History of branching processes 1
1.2 Single-type Galton-Watson branching process 2
1.3 Multitype Galton-Watson branching process 3

2 2-Periodic Multitype Branching Processes 9
2.1 Basic notation and definition 9
2.2 Limiting behaviors of the offspring mean matrices 11
2.3 Limits behaviors of population 15

3 Markov limit on types along the line of descent 19
3.1 Ancestral types along the line of descent 19
3.2 Markov properties on ancestral types 20

4 Conclusion 45

References 46

[1] K.B. Athreya and P.E. Ney. Branching processes. Springer,New York, 1972.
[2] Theodore Edward Harris et al. The theory of branching processes, volume 6. Springer Berlin, 1963.
[3] Edward Pollak. Survival probabilities and extinction times for some multitype branching processes. Advances in Applied Probability, 6(3):446–462, 1974.
[4] Jyy-I Hong and K.B. Athreya. Markov limit of line of decent types in a multitype supercritical branching process. Statistics & Probability Letters, 98:54–58, 2015.
[5] Jyy-I Hong. Coalescence on supercritical multi-type branching processes. Sankhya A, 77:65–78, 2015.
[6] Jyy-I Hong. Coalescence on critical and subcritical multitype branching processes. Journal of Applied Probability, 53(3):802–817, 2016.