The study of different modes of convergence for sequences of measures, including weak convergence, vague convergence, narrow convergence, and their interrelationships. Central to probability theory (convergence in distribution), functional analysis, and applications in statistics, mathematical finance, and PDE theory.
Hierarchy of Convergence Types
For finite signed measures on R, the following strict hierarchy holds (Stanek 2024):
weak ⇒ loose ⇒ vague ⇒ basic ⇒ almost basic
For non-negative measures, vague = basic = almost basic (the hierarchy collapses). The key equivalences governing the hierarchy are:
| Condition A | + | Condition B | ⇐> | Result |
|---|---|---|---|---|
| vague convergence | + | tightness | ⇐> | weak convergence |
| vague convergence | + | no mass lost on compact sets | ⇐> | vague convergence of H-J parts |
| vague convergence | + | lim sup ||mu_n|| ⇐ ||mu|| | ⇐> | weak convergence of H-J parts |
| basic or almost basic | + | local uniform boundedness in variation | ⇐> | vague convergence |
| basic or almost basic | + | uniform boundedness in variation | ⇐> | loose convergence |
Unifying Frameworks
Hu’s theory of boundedness (1966), applied by Basrak-Planinic (2018), shows that different choices of “bounded sets” yield different flavours of vague convergence — including classical Radon measure vague convergence, w#-convergence, and M_0-convergence in extreme value theory — all as special cases.
Papers Analyzed
- Portmanteau theorem for unbounded measures — extends the portmanteau theorem to Levy-type measures (Barczy-Pap 2006)
- A note on vague convergence of measures — unifies vague convergence via Hu’s boundedness (Basrak-Planinic 2018)
- Vague and weak convergence of signed measures — comprehensive vague-weak equivalence for signed measures (Herdegen-Liang-Shelley 2022)
- Effective weak and vague convergence of measures on the real line — computable analysis perspective (Rojas 2021)
- Vague and basic convergence of signed measures — complete characterisation via basic convergence (Stanek 2024)
- Vague convergence and method of moments for random metric measure spaces — vague convergence in Gromov-weak topology (Foutel-Rodier 2024)
- Probability on Submetric Spaces — sequential Prokhorov theorem on non-metrizable submetric spaces via CCSP and Skorokhod representation (Jakubowski 2023)
- The almost sure Skorokhod representation for subsequences in nonmetric spaces — foundational result: CCSP + uniform tightness gives Skorokhod representation for subsequences without metrizability (Jakubowski 1997)
Key Concepts
- vague convergence — test against C_c functions
- weak convergence of measures — test against C_b functions
- loose convergence — test against C_0 functions
- basic convergence — subsequential distribution function convergence
- almost basic convergence — relaxation to Lebesgue-null exceptional sets
- tightness — the bridge between vague and weak
- mass preserving condition — hierarchy controlling mass loss for signed measures
- portmanteau theorem — equivalent characterisations of convergence
- Prokhorov metric — metrises weak convergence
- effective convergence — computable analysis perspective
- Gromov-weak topology — convergence of metric measure spaces
- method of moments for metric measure spaces — moments of order k >= 1 suffice for vague convergence
Textbooks
- Measure Theory - Bogachev — Comprehensive treatment of weak convergence on general topological spaces, Prohorov’s theorem, Alexandroff’s portmanteau, signed measure convergence (Ch. 7.9-7.11, 8.1-8.9)
- Infinite Dimensional Analysis — Functional-analytic treatment: w*-topology on P(X), portmanteau for nets, Prokhorov’s theorem, AL-space structure of ba(A)/ca(A), Vitali-Hahn-Saks, regularity and tightness on Polish spaces (Ch. 10, 12, 15)
Open Questions
- Generalise effective weak and vague convergence from M(R) to M(X) for arbitrary computable metric spaces X
- Explore whether the Stanek characterisation (vague ⇐> basic + local boundedness) extends beyond R to general locally compact spaces
- Further applications of vague convergence in the Gromov-weak topology to spatial branching processes and population genetics