Cramer-Wold Theorem

The Cramer-Wold Theorem, also called the Cramer-Wold device, states that a Borel probability measure on k-dimensional Euclidean space is uniquely determined by the totality of its one-dimensional projections, so that a sequence of random vectors converges in distribution to a limiting random vector exactly when every fixed linear combination of the sequence's coordinates converges in distribution to the corresponding linear combination of the limit's coordinates. Named after Harald Cramer and Herman Wold, who published the result in 1936, it is used as a standard method for proving joint convergence results in probability theory.