Cramér–Wold theorem

Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on Rk 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

\overline{X}_n = (X_{n1},\dots,X_{nk}) \;

and

\; \overline{X} = (X_1,\dots,X_k)

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

\sum_{i=1}^k t_iX_{ni} \frac{D}{\overrightarrow{\infty}} \sum_{i=1}^k t_iX_i.

for each (t_1,\dots,t_k)\in \mathbb{R}^k That is if every fixed linear combination of the coordinates of \overline{X}_n converges in distribution to the correspondent linear combination of coordinates of \overline{X} .

This article incorporates material from Cramér-Wold theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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