Cramer-Wold's theorem

The Cramér-Wold theorem , also Cramér-Wold device called (by Harald Cramér and Herman Wold ) from the measure theory states that a Borel measure on is uniquely defined one-dimensional through all of its projections. This is the reason why it is sufficient in statistical procedures such as the Grand Tour or Projection Pursuit to look at projections of the data. It was published in 1936. ${\displaystyle \mathbb {R} ^{k}}$

Let there be and two -dimensional random variables . Then applies ${\displaystyle {\overline {X}}_{n}=(X_{n1},\dots ,X_{nk})\;}$ ${\displaystyle \;{\overline {X}}=(X_{1},\dots ,X_{k})}$ ${\displaystyle k}$

${\displaystyle {\overline {X}}_{n}{\rightarrow }{\overline {X}}\Longleftrightarrow \sum _{i=1}^{k}t_{i}X_{ni}{\rightarrow }\sum _{i=1}^{k}t_{i}X_{i}}$for everyone . ${\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} ^{k}}$

Every (fixed) linear combination of converges in distribution against the corresponding linear combination of if and only if converges against in distribution. This means that the convergence in the distribution of a multivariate random variable can be traced back to the convergence in the distribution of a univariate random variable (precisely the linear combinations). ${\displaystyle {\overline {X}}_{n}}$ ${\displaystyle {\overline {X}}}$ ${\displaystyle {\overline {X}}_{n}}$ ${\displaystyle {\overline {X}}}$

Web links

• PlanetMath: Cramér-Wold theorem

Individual evidence

1. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 335 , doi : 10.1007 / 978-3-642-36018-3 .
2. ^ Harald Cramér , Herman Wold : Some theorems on distribution functions. In: Journal of the London Mathematical Society. Serie 1, Vol. 11, No. 4, 1936, ISSN  0024-6107 , pp. 290-294, doi : 10.1112 / jlms / s1-11.4.290 .