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  "body": "\n## Statement\n\nLet $k \\ge 1$, and let $X, X_1, X_2, \\dots$ be random vectors in $\\R^k$.\nThen $X_n$ [[converges in distribution|0SRYJGXKR4]] to $X$ if and only if, for every $t \\in \\R^k$, the real random variables $t \\cdot X_n$ converge in distribution to $t \\cdot X$.\nHere $t \\cdot x$ is the [[dot product|0S52TNNTH9]] of $t$ and $x$.\n\n## Remarks\n\nProof idea: the [[characteristic function|0V5B6D6BQH]] of $X_n$ at $t$ is $E[e^{i t \\cdot X_n}]$, which is the characteristic function of $t \\cdot X_n$ at $1$.\nSo the condition gives pointwise convergence of the characteristic functions of $X_n$, and [[Lévy's continuity theorem|0FPWF7TTMB]] concludes.\nThe converse holds because $x \\mapsto t \\cdot x$ is continuous.\nA special case: the law of a random vector $X$ is determined by the laws of all its one-dimensional projections $t \\cdot X$.\nConvergence of the coordinates alone is not enough.\nIf $Z$ is standard normal, then $(Z, Z)$ and $(Z, -Z)$ have the same marginals but different laws.\nSo the constant sequence $(Z, Z)$ has coordinates that converge in law to those of $(Z, -Z)$, but it does not converge in law to $(Z, -Z)$.\n\n## Explanations\n\n### L1 human\n\nTo check that random vectors settle down to a limit law, it is enough to look at every straight-line projection of them.\nThis reduces questions about many dimensions to questions about one dimension.\n\n### L2 human\n\nThe standard use is the multivariate [[central limit theorem|0EPZC4Z1FC]]: for i.i.d. random vectors with finite covariance matrix, each projection $t \\cdot X_i$ obeys the one-dimensional theorem, and the Cramér–Wold theorem lifts this to a [[multivariate normal|0XN3K7RSZY]] limit.\nIt is also called the Cramér–Wold device.\n",
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