The Cramér–Wold theorem is a result in probability theory that provides a characterization of multivariate normal distributions. It states that a random vector follows a multivariate normal distribution if and only if every linear combination of its components is normally distributed. More formally, let \( X = (X_1, X_2, \ldots, X_n) \) be a random vector in \( \mathbb{R}^n \).
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Random vectors converge weakly to exactly when every linear projection converges weakly to . The reverse implication follows because these one-dimensional limits give pointwise convergence of the multivariate characteristic functions.