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| In [[mathematics]], the '''Cramér–Wold theorem''' in [[measure theory]] states that a [[Borel measure|Borel]] [[probability measure]] on <math>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]].
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| Let
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| :<math> \overline{X}_n = (X_{n1},\dots,X_{nk}) \; </math>
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| and
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| :<math> \; \overline{X} = (X_1,\dots,X_k) </math>
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| be [[random vector]]s of dimension ''k''. Then <math> \overline{X}_n </math> [[converges in distribution]] to <math> \overline{X} </math> if and only if:
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| :<math> \sum_{i=1}^k t_iX_{ni} \overset{D}{\underset{n\rightarrow\infty}{\rightarrow}} \sum_{i=1}^k t_iX_i. </math>
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| for each <math> (t_1,\dots,t_k)\in \mathbb{R}^k </math>, that is, if every fixed [[linear combination]] of the coordinates of <math> \overline{X}_n</math> converges in distribution to the correspondent linear combination of coordinates of <math> \overline{X} </math>.
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| {{PlanetMath attribution|id=3711|title=Cramér-Wold theorem}}
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| == External links ==
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| * Project Euclid: [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.aop/1024404418 "When is a probability measure determined by infinitely many projections?"]
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| {{DEFAULTSORT:Cramer-Wold theorem}}
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| [[Category:Theorems in measure theory]]
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| {{Mathanalysis-stub}}
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Revision as of 09:42, 14 February 2014
Hi there. Allow me begin by introducing the author, her name is Myrtle Cleary. Supervising is my occupation. To collect coins is what her family and her appreciate. For years he's been residing in North Dakota and his family members enjoys it.
Here is my homepage :: over the counter std test