A sequence of probability measures (μ_n) on a metric space converges weakly to μ if ∫ f dμ_n → ∫ f dμ for every bounded continuous function f; equivalently, distribution functions converge at continuity points of the limit.

Principle

Convergence is tested by bounded continuous observables rather than by pointwise or total-variation norms, capturing convergence in distribution of random elements.

Demonstration

Empirical measures built from i.i.d. samples converge weakly to the true underlying distribution almost surely by the Glivenko–Cantelli/various law-type results; e.g., normalized sums converge in law to a limiting stable law under appropriate conditions.

Misapplication

Using weak convergence to justify convergence of expectations of unbounded functions or concluding convergence in stronger metrics such as total variation without additional conditions.

Consequence

Guarantees convergence of integrals of bounded continuous test functions and of cumulative distribution functions at continuity points; under tightness, sequences admit subsequences with weak limits.

Reversal

Convergence in total variation or uniform convergence of densities, which implies stronger control and convergence of integrals of unbounded functions but is not implied by weak convergence.

Boundary

Defined for measures on metric spaces; does not imply convergence of moments unless moment conditions hold and is weaker than convergence in stronger norms.

Semantic Tension

Often contrasted with convergence in probability or almost-sure convergence of random variables; weak convergence pertains to laws rather than sample-path behavior.

Synthesis

A mode of convergence for measures capturing convergence of distributions via bounded continuous tests, central to limit theorems and probabilistic approximation.