Lévy’s continuity theorem

E1020437

result in measure-theoretic probability theorem in probability theory

Lévy’s continuity theorem is a fundamental result in probability theory that characterizes convergence in distribution of random variables via pointwise convergence of their characteristic functions.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (1)

Label	Occurrences
Lévy’s continuity theorem canonical	1

Statements (45)

Predicate	Object
instanceOf	result in measure-theoretic probability ⓘ theorem in probability theory ⓘ
alternativeName	continuity theorem for characteristic functions ⓘ
appearsIn	graduate-level probability textbooks ⓘ measure-theoretic treatments of probability ⓘ
appliesTo	random vectors in ℝ^d ⓘ real-valued random variables ⓘ
assumes	existence of characteristic functions for all measures ⓘ sequence of probability measures on ℝ^d ⓘ tightness is implied by convergence of characteristic functions with continuity at 0 ⓘ
characterizes	convergence in distribution ⓘ weak convergence of probability measures ⓘ
concerns	Fourier transforms of probability distributions ⓘ weak topology on space of probability measures ⓘ
concludes	weak convergence if characteristic functions converge pointwise and limit is continuous at 0 ⓘ
equivalenceBetween	pointwise convergence of characteristic functions on ℝ ⓘ weak convergence of associated probability measures ⓘ
field	measure theory ⓘ probability theory ⓘ
generalizedTo	locally compact abelian groups ⓘ
hasFormulation	if μ_n ⇒ μ then φ_n(t) → φ(t) for all t ⓘ if φ_n(t) → φ(t) for all t and φ is continuous at 0, then μ_n ⇒ μ ⓘ
holdsIn	Euclidean spaces ℝ^d NERFINISHED ⓘ
implies	uniqueness of probability measure determined by its characteristic function ⓘ
importance	connects analytic properties of characteristic functions with probabilistic convergence ⓘ fundamental tool for analyzing convergence of distributions ⓘ
isUsedFor	establishing convergence of stochastic processes in distribution ⓘ proving Donsker’s theorem ⓘ proving central limit theorems ⓘ proving functional central limit theorems ⓘ proving invariance principles ⓘ studying limit distributions of sums of independent random variables ⓘ
namedAfter	Paul Lévy NERFINISHED ⓘ
relatedTo	Bochner’s theorem NERFINISHED ⓘ Helly–Bray theorem NERFINISHED ⓘ Lévy–Khintchine formula NERFINISHED ⓘ Portmanteau theorem NERFINISHED ⓘ
relates	convergence in distribution of random variables ⓘ pointwise convergence of characteristic functions ⓘ
requiresCondition	continuity at 0 of the pointwise limit of characteristic functions ⓘ pointwise convergence of characteristic functions at every real argument ⓘ
typeOfConvergence	convergence in law ⓘ weak convergence ⓘ
usesConcept	Fourier transform of probability measures ⓘ characteristic function ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Paul Lévy → knownFor → Lévy’s continuity theorem ⓘ