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 |
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Herman Wold NERFINISHED ⓘ |
| relatedTo |
Skorokhod representation theorem
NERFINISHED
ⓘ
central limit theorem NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.