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研究生： 劉丞邦
Liu, Cheng-Pang
論文名稱： 生滅鏈的相變現象
Total variation cutoffs for birth and death chains
指導教授： 陳冠宇
Chen, Guan-Yu
鄭志豪
Teh, Jyh-Haur
口試委員: 陳隆奇
Chen, Lung-Chi
須上苑
Shiu, Shang-Yuan
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 29
中文關鍵詞： 馬可夫鏈 、生滅鏈 、相變現象
外文關鍵詞： Markov chains, birth and death chains, cutoff phenomenon, cutoffs, total variation cutoffs
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    • 馬可夫鏈的相變現象是指其於收斂至穩態分布的過程中，其與穩態分布間的距離發生劇烈的轉變。對於從邊界出發的生滅鏈，Chen 和 Saloff-Coste (2015)證明了相變現象等價於擊中時間具有某種機率分布集中的性質，而這篇論文考慮的生滅鏈，則是從穩態分布觀點下的邊界附近出發，我們發現擊中時間具有這種機率分布集中的性質，對於相變現象是一個充分但非必要的條件。

    The cutoff phenomenon describes a case where a Markov chain converges abruptly to stationarity. For birth and death chains starting from boundary, Chen and Saloff-Coste (2015) proved that the occurrence of a total variation cutoff is equivalent to some kind of measure concentration of hitting times. This thesis considers birth and death chains starting from, in the sense of stationary distributions, near boundary and shows that this kind of measure concentration of hitting times is sufficient but not necessary for a total variation cutoff to occur.

    1 Introduction. . . . . . . . . .. . . . . . . . . . . . . . . . 1 2 Terminology. . . . . . . . . .. . . . . . . . . . . . . . . . .3 2.1 Distances and mixing times . . . . . . . . . . . . . . . . 3 2.2 Cutoff phenomenon . . . . . . . . . . . . . . . . . . . . 6 2.3 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Birth and death chains. . . .. . . . . . . . . . . . . . . . 11 3.1 Bound the total variation distance by couplings . . . . . 12 3.2 Expectations and variances of hitting times . . . . . . . 15 3.2.1 Form of Spectra . . . . . . . . . . . . . . . . . . . 15 3.2.2 Form of transition rates . . . . . . . . . . . . . . .18 4 Main theorem. . . . . . . . .. . . . . . . . . . . . . . . . 21 5 Ongoing works. . . . . . . . .. . . . . . . . . . . . . . . . 25 A Auxiliary results and proofs. . . . . . . . .. . . . . . . . 27

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