Cramér-Wold theorem

In mathematics, the Cramér-Wold theorem in measure theory states that a Borel probability measure on $$R^k is uniquely determined by the totality of its one-dimensional projections. 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 $$overline{X}_n converges to $$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} .

External links

* Project Euclid: [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.aop/1024404418 "When is a probability measure determined by infinitely many projections?"]
* Reference.com: [http://www.reference.com/browse/wiki/Herman_Wold "Herman Wold"]