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. 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:

$\sum_{i=1}^k t_iX_{ni} \overset{D}{\underset{n\rightarrow\infty}{\rightarrow}} \sum_{i=1}^k t_iX_i.$

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 $\overline{X}$.

Proof

For a proof, see for example Billingsley (1995, p. 383)

References

• This article incorporates material from Cramér-Wold theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Billingsley, Patrick (1995). Probability and Measure. John Wiley. ISBN 978-0-471-00710-4.
• Cramér, Harald; Wold, Herman (1936). "Some Theorems on Distribution Functions". Journal of the London Mathematical Society 11 (4): 290–294. doi:10.1112/jlms/s1-11.4.290.

External links

• Project Euclid: "When is a probability measure determined by infinitely many projections?"