Cramér–Wold theorem
In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on [math]\displaystyle{ \mathbb{R}^k }[/math] 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
- [math]\displaystyle{ {X}_n = (X_{n1},\dots,X_{nk}) }[/math]
and
- [math]\displaystyle{ \; {X} = (X_1,\dots,X_k) }[/math]
be random vectors of dimension k. Then [math]\displaystyle{ {X}_n }[/math] converges in distribution to [math]\displaystyle{ {X} }[/math] if and only if:
- [math]\displaystyle{ \sum_{i=1}^k t_iX_{ni} \overset{D}{\underset{n\rightarrow\infty}{\rightarrow}} \sum_{i=1}^k t_iX_i. }[/math]
for each [math]\displaystyle{ (t_1,\dots,t_k)\in \mathbb{R}^k }[/math], that is, if every fixed linear combination of the coordinates of [math]\displaystyle{ {X}_n }[/math] converges in distribution to the correspondent linear combination of coordinates of [math]\displaystyle{ {X} }[/math].[1]
If [math]\displaystyle{ {X}_n }[/math] takes values in [math]\displaystyle{ \mathbb{R}_+^k }[/math], then the statement is also true with [math]\displaystyle{ (t_1,\dots,t_k)\in \mathbb{R}_+^k }[/math].[2]
Footnotes
- ↑ Billingsley 1995, p. 383
- ↑ Kallenberg, Olav (2002). Foundations of modern probability (2nd ed.). New York: Springer. ISBN 0-387-94957-7. OCLC 46937587. https://www.worldcat.org/oclc/46937587.
References
- 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
 |  |
