Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on ${\displaystyle \mathbb {R} ^{k}}$ is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.

Let

and

be random vectors of dimension k. Then ${\displaystyle {\overline {X}}_{n}}$ converges in distribution to ${\displaystyle {\overline {X}}}$ if and only if:

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