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"]