Weak convergence of borel probability measures

Depends on

• Probability measures and probability spaces
• The Borel sigma-algebra of a topological space
• Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
• Integrable real and complex functions, and their integrals

Used by

• Deterministic Ito integrals are Gaussian Corollary
• Fixed negative-axis directional resolvent integrals converge Corollary
• Isotropic minimum-norm risk limits under a positive spectral gap Corollary
• Studentized RQMC replicate means are asymptotically normal as R→∞ Corollary
• The scaled posterior converges weakly on the negative Sobolev space Corollary
• 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
• Asymptotic distribution Definition
• Conditional weak convergence in probability Definition
• Convergence in distribution of random elements Definition
• Relative sequential compactness for weak convergence Definition
• Sample covariance matrices and their empirical spectral distributions Definition
• A degenerate weak limit with persistent expectation Example
• A singular Brownian data law forces positive-time score truncation Example
• At aspect one the reciprocal-spectrum limit diverges Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• 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
• Weakly close root laws can still give failed uniform coverage Example
• Weakly convergent finite-step output laws need not be exact Example
• Compact capacity-zero sets and subharmonic minus-infinity loci Lemma
• Countable compactly supported tests determine euclidean weak convergence Lemma
• Finite-horizon bootstrap particle filters are weakly consistent Lemma
• Probability laws on a compact metric space have weakly convergent subsequences Lemma
• Relative property (T) for SL2(R) semidirect R2 Lemma
• The bounded-Lipschitz metric characterizes weak convergence on Polish spaces Lemma
• Optimal quadratic couplings exist and their cost is weakly lower semicontinuous Proposition
• Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices Theorem
• Conditional K-sample block-bootstrap validity under a joint CLT Theorem
• Cramer wold device Theorem
• Directional deterministic equivalent for bounded sample-covariance entries Theorem
• Existence and uniqueness of the equilibrium measure Theorem
• Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant Theorem
• Fixed-horizon particle backward smoothing is consistent under stated conditions Theorem
• Gaussian smoothing converges weakly at the endpoint Theorem
• Gaussian smoothing density and forward equation Theorem

…and 13 more results.

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)