Cramér–Wold theorem
E933486
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cramér–Wold theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11560422 — 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: Cramér–Wold theorem Context triple: [Harald Cramér, knownFor, Cramér–Wold theorem]
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A.
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.
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B.
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.
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C.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
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D.
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.
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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: Cramér–Wold theorem Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
D.
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.
-
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 (30)
| Predicate | Object |
|---|---|
| instanceOf |
result in multivariate statistics
ⓘ
theorem in probability theory ⓘ |
| appliesTo |
multivariate probability distributions
ⓘ
random vectors in R^n ⓘ |
| category |
theorems in probability theory
ⓘ
theorems in statistics ⓘ |
| conclusion | two random vectors with identical distributions of all linear projections have the same multivariate distribution ⓘ |
| condition |
all one-dimensional linear projections must be considered
ⓘ
equality in distribution of all linear projections implies equality in distribution of random vectors ⓘ |
| describes | characterization of multivariate distributions by one-dimensional projections ⓘ |
| domain |
Euclidean space R^n
ⓘ
linked to:
Euclidean space
|
| field |
probability theory
ⓘ
statistics ⓘ |
| implies | uniqueness of a multivariate distribution from all one-dimensional linear projections ⓘ |
| involves |
linear combinations of components of a random vector
ⓘ
one-dimensional marginal distributions of linear projections ⓘ |
| mathematicalForm | If a^T X and a^T Y have the same distribution for all a in R^n, then X and Y have the same distribution ⓘ |
| namedAfter |
Harald Cramér
ⓘ
Herman Wold ⓘ |
| relatedTo |
Skorokhod representation theorem
ⓘ
central limit theorem ⓘ characteristic functions in probability theory ⓘ weak convergence of probability measures ⓘ |
| statement | A probability distribution on R^n is uniquely determined by the distributions of all its one-dimensional linear projections ⓘ |
| usedFor |
characterizing weak convergence in R^n
ⓘ
proving convergence in distribution of random vectors ⓘ reducing multivariate distribution problems to univariate ones ⓘ |
| usedIn |
asymptotic theory in statistics
ⓘ
multivariate central limit theorem proofs ⓘ theory of random vectors ⓘ |
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: Cramér–Wold theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.