Lévy’s continuity theorem
E1020437
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lévy continuity theorem | 1 |
| Lévy’s continuity theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T13070795 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lévy’s continuity theorem Context triple: [Paul Lévy, knownFor, Lévy’s continuity theorem]
-
A.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
-
B.
Cramér–Wold theorem
The Cramér–Wold theorem is a fundamental result in probability theory stating that a multivariate distribution is uniquely determined by the distributions of all its one-dimensional linear projections.
-
C.
Bochner theorem on characteristic functions
The Bochner theorem on characteristic functions is a fundamental result in probability theory and harmonic analysis that characterizes which functions are Fourier transforms of probability measures by requiring them to be positive-definite, continuous, and normalized at zero.
-
D.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lévy’s continuity theorem Target entity description: 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.
-
A.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
-
B.
Cramér–Wold theorem
The Cramér–Wold theorem is a fundamental result in probability theory stating that a multivariate distribution is uniquely determined by the distributions of all its one-dimensional linear projections.
-
C.
Bochner theorem on characteristic functions
The Bochner theorem on characteristic functions is a fundamental result in probability theory and harmonic analysis that characterizes which functions are Fourier transforms of probability measures by requiring them to be positive-definite, continuous, and normalized at zero.
-
D.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- F. None of above. chosen
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
ⓘ
linked to:
Euclidean space
|
| 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 ⓘ |
| relatedTo |
Bochner’s theorem
ⓘ
linked to:
Bochner theorem on characteristic functions
Helly–Bray theorem ⓘ
linked to:
Prokhorov’s theorem
Lévy–Khintchine formula ⓘ Portmanteau theorem ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lévy’s continuity theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.