Cramér–Wold theorem

The Cramér–Wold theorem states that two probability distributions on Euclidean space are equal if all their one-dimensional linear projections have the same distribution.

Pushforward measure: A measure transported through a measurable map by assigning each output set the measure of its preimage. Each linear projection transports a multivariate distribution to a one-dimensional distribution.
Probability distribution: A mathematical description of the possible outcomes of a random quantity and their probabilities. The theorem compares probability distributions rather than individual random vectors.
Multivariate central limit theorem: The theorem that normalized sums of random vectors converge to a multivariate normal distribution under suitable conditions. Scalar central limit results for all linear combinations can establish the vector-valued conclusion.
Copula: A function that describes the dependence structure joining specified marginal distributions. Matching coordinate marginals does not determine a joint law; the theorem requires projections in all directions.
Characteristic function: A function that encodes a probability distribution through the expected value of complex exponentials. Projection laws determine the multivariate characteristic function along every line through the origin.
Random vector: A random quantity whose value is a vector of real-valued random variables. The theorem concerns the joint distribution represented by a random vector.
Cramér–Wold device for weak convergence: A criterion that proves weak convergence of random vectors by proving convergence of every scalar linear combination. It is the theorem’s best-known application in asymptotic probability.
Joint distribution: The probability distribution of a collection of random variables considered together. The theorem identifies this full dependence structure from all scalar projections, not merely separate coordinates.
Fourier transform of a probability measure: The Fourier transform of a probability measure, evaluated as an expectation of complex exponentials. The theorem’s proof uses Fourier transforms to recover the joint law from directional information.
Linear functional: A linear map from a vector space to its scalar field. Every one-dimensional projection in the theorem is a linear functional on Euclidean space.

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Cramér's decomposition theoremCompared with: It shares Cramér's name but identifies multivariate distributions rather than summands.