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

${\displaystyle {X}_{n}=(X_{n1},\dots ,X_{nk})}$

and

${\displaystyle \;{X}=(X_{1},\dots ,X_{k})}$

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

${\displaystyle \sum _{i=1}^{k}t_{i}X_{ni}{\overset {D}{\underset {n\rightarrow \infty }{\rightarrow }}}\sum _{i=1}^{k}t_{i}X_{i}.}$

for each ${\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} ^{k}}$, that is, if every fixed linear combination of the coordinates of ${\displaystyle {X}_{n}}$ converges in distribution to the correspondent linear combination of coordinates of ${\displaystyle {X}}$.^[1]

If ${\displaystyle {X}_{n}}$ takes values in ${\displaystyle \mathbb {R} _{+}^{k}}$, then the statement is also true with ${\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} _{+}^{k}}$.^[2]

Footnotes

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 (3 ed.). John Wiley & Sons. 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?"