A note on vague convergence of measures

TL;DR

This paper introduces a unified approach to vague convergence of measures using Hu's boundedness theory, enabling translation of results across different types of convergence and providing new insights into convergence in distribution of random measures.

Contribution

It presents a novel framework connecting various types of vague convergence, simplifying the transfer of results and offering a new characterization of convergence in distribution.

Findings

Unified framework for vague convergence based on boundedness

Connection and unification of different vague convergence types

New characterization of convergence in distribution of random measures

Abstract

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.