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| 題名: | 多型態分支隨機漫步的位置分佈
Position distributions in multi-type branching random walks |
| 作者: | 許書睿
Hsu, Shu-Jui |
| 貢獻者: | 洪芷漪
Hong, Jyy-I
許書睿
Hsu, Shu-Jui |
| 關鍵詞: | 分支過程
多型態
分支隨機漫步
溯祖問題
位置分佈
超臨界
經驗分佈
Branching process
Multi-type
Branching random walks
Coalescence problem
Position problem
Supercritical
Empirical distribution |
| 日期: | 2025 |
| 上傳時間: | 2025-07-01 14:41:07 (UTC+8) |
| 摘要: | 我們研究具有 L 種型態的超臨界多型態分支過程 {Z_n}_{n≥0}。首先，在適當的條件下，探討從第 n 代隨機選取一個個體時，其祖先型態的漸近比例。接著，我們考慮此過程在實數線 R 上所對應的分支隨機漫步。令 Z_{n,i}(x) 表示第 n 代中，位置小於或等於 x 的型態 i 個體數。我們證明存在一發散的數列 {β_n}_{n=0}^∞，使得 Z_{n,i}(β_nx)/|Zn| 以 L^2 的形式收斂。最後，我們證明從第 n代中隨機選取一個個體，其位置在機率分布上收斂至標準常態分布。
We study a supercritical multi-type branching process {Z_n}_{n≥0} with L types. First, we investigate, under suitable conditions, the asymptotic proportion of ancestral types for an individual randomly selected from generation n. Next, we consider the associated branching random walk on the real line R. Let Z_{n,i}(x) denote the number of type-i individuals in generation n whose positions are less than or equal to x. We show that there is a sequence {β_n}_{n=0}^∞ with β_n → ∞ such that the ratio Z_{n,i}(β_nx)/|Zn| converges in L^2. Finally, we establish that the position of a uniformly chosen individual from generation n converges in distribution to the standard normal law. |
| 參考文獻: | [1] K.B. Athreya. Discounted branching random walks. Advances in applied probability, 17(1):53–66, 1985.
[2] K.B. Athreya. Branching random walks. The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pages 337–349, 2010.
[3] K.B. Athreya. Coalescence in the recent past in rapidly growing populations. Stochastic Processes and their Applications, 122(11):3757–3766, 2012.
[4] K.B. Athreya and J.-I. Hong. An application of the coalescence theory to branching random walks. Journal of Applied Probability, 50(3):893–899, 2013.
[5] K.B. Athreya and J.-I. Hong. Markov limit of line of decent types in a multitype supercritical branching process. Statistics & Probability Letters, 98:54–58, 2015.
[6] K.B. Athreya and P. E. Ney. Branching processes. Courier Corporation, 2004.
[7] J.-L. Chi and J.-I. Hong. The range of asymmetric branching random walk. Statistics & Probability Letters, 193:109705, 2023.
[8] P.L. Davies. The simple branching process: a note on convergence when the mean is infinite. Journal of Applied Probability, 15(3):466–480, 1978.
[9] R. Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019.
[10] F. Galton and H. W. Watson. On the probability of the extinction of families. The Journal of the Anthropological Institute of Great Britain and Ireland, 4:138–144, 1875.
[11] D.R. Grey. Almost sure convergence in markov branching processes with infinite mean. Journal of Applied Probability, 14(4):702–716, 1977.
[12] J.-I. Hong. Coalescence on supercritical multi-type branching processes. Sankhya A, 77:65–78, 2015.
[13] J.-I. Hong. Coalescence on critical and subcritical multi-type branching processes. Journal of Applied Probability, 53(3):802–817, 2016.
[14] W. Yang and Z. Ye. The asymptotic equipartition property for non-homogeneous Markov chains indexed by a homogeneous tree. IEEE Transactions on Information Theory, 53(9):3275–3280, 2007. |
| 描述: | 碩士
國立政治大學
應用數學系
112751001 |
| 資料來源: | http://thesis.lib.nccu.edu.tw/record/#G0112751001 |
| 資料類型: | thesis |
| 顯示於類別: | [應用數學系] 學位論文
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