Stein 方法與其在最小生成樹上的應用

Abstract

這篇論文主要分成兩大部分: 第一章介紹 Stein 方法的歷史與理論；第二、三、四章則著重於該方法在最小生成樹上的具體應用。在第一章中我們分別介紹了兩種不同用途的 Stein 方法，分別用於中央極限定理與集中不等式。在應用層面，我們採用 [4] 中Chatterjee 提出的框架去證明最小生成樹上的中央極限定理。在集中不等式方面，我們推廣了 [2] 中 Chatterjee 提出的方法，並將其稱作廣義擾動法。同時，在最後一章中，我們分別使用廣義擾動法與 [2] 中的方法去獲得 Zd中最小生成樹的集中不等式並簡單探討兩者的優劣。

This thesis is primarily composed of two parts: the first is an introduction to Stein’s method and the other focuses on the applications of Stein’s method to the minimal spanning trees (MSTs).In Chapter 1, we introduce two versions of Stein’s method for different purposes: deriving a central limit theorem and establishing concentration inequalities.In the subsequent chapters, we use Stein’s method to prove a central limit theorem and concentration inequalities for the MSTs.For the central limit theorem, we use the approach introduced by Chatterjee in [4]. Regarding concentration inequalities, we generalize the approach introduced by Chatterjee in [2], which we refer to as the generalized perturbative method. Finally, in the last chapter, we provide both the original method from [2] and the generalized perturbative method to derive concentration inequalities for the MSTs, concluding with a brief discussion on the advantages and disadvantages of these two approaches.