Abstract
We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.
Summary
This paper extends the classical portmanteau theorem — originally a characterisation of weak convergence for probability measures — to measures that may be unbounded, provided they are finite on complements of neighbourhoods of a fixed element x_0. The key setting is a metric space (X, d) where measures eta_n satisfy eta_n(X \ U) < infinity for all Borel neighbourhoods U of x_0.
The main result (Theorem 2.1) establishes the equivalence of six conditions for convergence of such unbounded measures, including: convergence of integrals against continuous functions in C(X), C_{x_0}(X), and BL_{x_0}(X); weak convergence of restrictions eta_n|_{X\U}; convergence of measures of complements eta_n(X\U); and a portmanteau-style lim sup / lim inf condition on open and closed neighbourhoods. The proof follows the classical portmanteau argument but must be adapted since the measures are no longer bounded on X.
A concrete motivation is Levy measures in the theory of infinitely divisible distributions, where the classical portmanteau theorem does not directly apply because Levy measures are unbounded at the origin.
Key Contributions
- Extends the portmanteau theorem to unbounded measures finite away from a fixed point
- Corrects errors in Propositions 1.2.13 and 1.2.19 of Meerschaert and Scheffler (counterexamples in Remarks 2.3 and 2.4 show conditions (c) and (d) are not equivalent)
- Removes the separability assumption on the underlying metric space
- Provides six equivalent characterisations of convergence for this class of measures
Methodology
The proof strategy follows the classical portmanteau argument (Dudley’s Theorem 11.1.1) but replaces bounded continuous functions with functions vanishing near x_0. The key innovation is constructing appropriate Lipschitz cutoff functions that separate sets away from x_0, allowing the classical proof structure to carry through for unbounded measures.
Key Findings
- The six conditions (i)-(vi) in Theorem 2.1 are equivalent for measures finite on complements of neighbourhoods of x_0
- Conditions (a) and (b) in condition (vi) — the lim sup on open sets and lim inf on closed sets — are NOT individually equivalent to the other conditions (Remark 2.3 provides a counterexample)
- The result applies directly to convergence of Levy measures in the theory of infinitely divisible probability distributions
Important References
- Real Analysis and Probability — Dudley’s textbook containing the classical portmanteau theorem
- Limit Distributions for Sums of Independent Random Vectors — Meerschaert and Scheffler’s treatment of portmanteau for bounded/unbounded measures
- Limit Theorems for Stochastic Processes — Jacod and Shiryayev’s conditions for weak convergence of infinitely divisible measures