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Weak convergence of borel probability measures
Definition
For Borel probability measures on a metric space S, write if for every bounded continuous real function f on S. Continuity is Continuity of a map between metric spaces, at a point and globally, in the - form. Such f is Borel measurable (inverse images of open sets are open) and , so the integrals are finite in Integrable real and complex functions, and their integrals. No completeness or coupling is required.
Depends on
Used by
- Bounded continuous cannot be replaced by all bounded measurable functions Counterexample
- Boundedness of first moments alone does not give uniform integrability Counterexample
- Pointwise cdf convergence at a jump is not required Counterexample
- Convergence in distribution of random elements Definition
- Relative sequential compactness for weak convergence Definition
- Dirac laws converge weakly exactly when their points converge Example
- Empirical laws of a finite valued iid sample Example
- Uniform laws on expanding finite grids converge to uniform zero one Example
- Weak convergence of gaussian laws by parameters Example
- Countable compactly supported tests determine euclidean weak convergence Lemma
- Probability laws on a compact metric space have weakly convergent subsequences Lemma
- Cramer wold device Theorem
- Levy continuity theorem converse Theorem
- Levy continuity theorem forward direction Theorem
- Portmanteau theorem Theorem
Dependency tree · two levels
15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- van Gaans, Definition 3.1, pp. 6–7 (standard reference, not scraped)