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.
Summary
This paper provides a unifying framework for understanding the many different notions of vague convergence that appear throughout the literature, by grounding them all in Hu’s 1966 abstract theory of boundedness. A family B_b of Borel subsets of a Polish space X is called a boundedness if it satisfies closure under subsets, finite unions, and has a basis with countable cover. Choosing different families B_b yields different notions of “locally finite” measures and hence different notions of vague convergence.
The key insight is that the choice of bounded sets B_b completely determines which measures are considered “locally finite” (those finite on all B in B_b) and which test functions are used for vague convergence (those with support in some bounded set). When B_b is the family of all Borel sets, one recovers weak convergence of finite measures. When B_b is the family of relatively compact Borel sets (in a locally compact space), one recovers the classical vague convergence of Radon measures. When B_b consists of sets bounded away from a closed set C, one recovers the M_0-convergence / Hult-Lindskog convergence used in extreme value theory.
The paper further proves that the vague topology on M(X) is metrizable and Polish (Proposition 3.1), constructing an explicit metric via projection maps T_m that truncate measures to increasingly large bounded sets. Section 4 gives sufficient conditions for convergence in distribution of random measures in the vague topology via Laplace functionals.
Key Contributions
- Unifies weak convergence, classical vague convergence, w#-convergence, and Hult-Lindskog M_0-convergence as special cases of a single framework
- Shows vague convergence is equivalent to convergence on continuity sets (Portmanteau theorem follows from Kallenberg)
- Proves the vague topology is metrizable and Polish, providing an explicit metric
- Gives Laplace functional characterisation of convergence in distribution of random measures
- Extends Lohr-Rippl’s Stone-Weierstrass-based convergence determining results to random measures
Methodology
The approach builds on Hu’s theory of boundedness to define a “proper localising sequence” (K_m) — a sequence of open bounded sets covering X with nested closures. The vague topology is then defined via the family of continuous bounded functions with support in some bounded set. Metrizability is established by constructing truncation maps T_m: M(X) → M-hat(X) using cutoff functions g_m, and defining a metric as a weighted sum of Prohorov distances between truncated measures.
Key Findings
- Different choices of bounded families B_b yield different flavours of vague convergence — all within one unified theory
- The Portmanteau theorem (convergence on continuity sets iff vague convergence) holds automatically in this framework
- Vague convergence of point measures characterised by convergence of atom locations in bounded sets (Proposition 2.8)
- Convergence in distribution of random measures is equivalent to convergence of Laplace functionals for Lipschitz continuous functions with bounded support (Proposition 4.1)
- Lohr-Rippl’s Stone-Weierstrass convergence-determining results extend to random measures (Proposition 4.5)
Important References
- Random measures, theory and applications — Kallenberg’s comprehensive treatment of vague topology
- Introduction to General Topology — Hu’s 1966 foundational work on abstract boundedness
- Extreme values, regular variation and point processes — Resnick’s approach to vague convergence for point processes