Portmanteau定理的一个自包含证明
A Self-Contained Proof of the Portmanteau Theorem
- Department of Statistical Sciences, University of Toronto（多伦多大学统计科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文以七条陈述形式统一证明Portmanteau定理，强调有界连续函数期望与开闭集概率界的蕴含循环，并阐明中间函数类的作用。
AI中文摘要：
Portmanteau定理给出了概率测度弱收敛（等价地，随机变量依分布收敛）的若干等价刻画。本文以七条陈述的形式给出了该定理的统一证明，强调了连接有界连续函数期望与开集和闭集上概率界之间的蕴含循环结构。论证突出了标准逼近技术，并阐明了有界Lipschitz函数等中间函数类的作用。
英文摘要：
The Portmanteau Theorem gives several equivalent characterizations of weak convergence of probability measures (equivalently, convergence in distribution of random variables). This note presents a unified proof of the theorem in a seven-statement formulation, emphasizing the structure of the implication cycle connecting expectations of bounded continuous functions with probability bounds on open and closed sets. The argument highlights standard approximation techniques and clarifies the role of intermediate function classes such as bounded Lipschitz functions.
