Connection
For positive measures on R, vague convergence is equivalent to pointwise convergence of distribution functions at continuity points of the limit (Herdegen-Liang-Shelley, Theorem 3.2). For signed measures, this equivalence fails: the positive and negative parts of the Hahn-Jordan decomposition can cancel locally, causing distribution functions to diverge even when the measures converge vaguely (Examples 3.4 and 3.5 in Herdegen-Liang-Shelley). Their solution was to impose the “no mass at any point” condition (Definition 3.6), which rescues the equivalence (Theorem 3.8/3.11).
Stanek (2024) introduced basic convergence and proved the definitive characterisation: vague convergence of signed measures is equivalent to basic (or almost basic) convergence plus local uniform boundedness in variation (Theorem 3.12) — with NO mass condition needed. What has not been explicitly observed is that basic convergence is the precise concept that Herdegen-Liang-Shelley’s theory was “missing”: it is the minimal generalisation of pointwise distribution function convergence that drops the no-mass condition entirely.
The key insight is this: when a sequence {mu_n} has no mass at any point x in R (in the sense of Definition 3.6 in Herdegen-Liang-Shelley), then basic convergence reduces exactly to pointwise convergence of distribution function differences at all but countably many points. The “no mass” condition ensures that the exceptional countable set S in the definition of basic convergence can only contain points where the limit measure has atoms — and when there are no atoms, S is empty modulo continuity points. So the Herdegen-Liang-Shelley condition is precisely the condition under which the two notions coincide, and basic convergence is the strictly more general object that works without it.
Bridged Concepts
From Vague and weak convergence of signed measures
- mass preserving condition: The “local (zero) mass preserving condition” (Definition 3.6) was introduced to rescue the equivalence between vague convergence and distribution function convergence. It prevents positive and negative mass from cancelling undetectably.
- vague convergence ←> distribution function convergence: Theorem 3.8 gives the equivalence but needs the no-mass condition; Theorem 3.11 strengthens it under no mass at any point.
From Vague and basic convergence of signed measures
- basic convergence: Replaces pointwise convergence of F(x) - F(y) with subsequential convergence outside a countable set. This captures “convergence of distribution functions up to the right exceptional set.”
- Theorem 3.12: The complete characterisation vague ⇐> basic + local boundedness, without any mass condition.
- Theorem 3.15: Stanek restates Herdegen-Liang-Shelley’s Theorem 3.8 in his framework, explicitly noting the comparison. But he does not state the meta-observation that basic convergence is the canonical generalisation that subsumes the no-mass approach.
Why It Matters
This observation has direct relevance for the paper A Tauberian Theorem for signed measures (Herdegen, Liang, Shelley), which was the original motivation for studying vague convergence of signed measures. The Tauberian theorem requires passing between vague convergence and distribution function convergence. Currently, this passage uses the no-mass condition. Replacing pointwise convergence with basic convergence would:
- Remove the no-mass assumption from the Tauberian theorem, broadening its applicability to signed measures with atoms (which arise naturally in stochastic control problems with discrete state spaces or discrete optimal stopping times).
- Simplify the logical structure: instead of “vague convergence + no mass ⇒ distribution function convergence ⇒ desired conclusion,” one would have “vague convergence ⇒ basic convergence (automatic via Theorem 3.12) ⇒ desired conclusion.”
- Clarify the mathematical narrative: basic convergence is the natural language for distribution function convergence of signed measures, just as pointwise convergence is for positive measures.
Potential Directions
- Restate the Tauberian theorem of Herdegen-Liang-Shelley using basic convergence in place of pointwise distribution function convergence, thereby removing the no-mass hypothesis
- Investigate whether the Karamata-type asymptotic analysis in the Tauberian theorem can be carried out directly in terms of basic convergence, bypassing distribution functions entirely
- Characterise which classical results in probability theory that use “convergence of distribution functions at continuity points” can be generalised to signed measures by replacing with basic convergence
Evidence
- Stanek, Theorem 3.15 explicitly restates Herdegen-Liang-Shelley’s Theorem 3.8 and notes that basic convergence replaces pointwise convergence, dropping the no-mass condition
- Stanek, Remark 3.17(c): for non-negative measures, basic = almost basic = vague, recovering the classical theory as a special case
- Herdegen-Liang-Shelley, Section 3: the elaborate development of the no-mass condition and its counterexamples demonstrates the difficulty that basic convergence resolves
- A Tauberian Theorem for signed measures (arXiv:2205.13075): the companion paper whose Tauberian condition could potentially be weakened using basic convergence
Suggested Papers
- On weak convergence of quasi-infinitely divisible laws — Khartov’s original introduction of basic convergence, which would clarify the precise conditions under which basic convergence was designed to operate
- Weak convergence of measures — Bogachev’s monograph, which Stanek uses as a primary reference for the classical theory