A Self-Contained Proof of the Portmanteau Theorem

Source abstract

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.