PlanetMath: Cramér-Wold theorem

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Cramér-Wold theorem

(Theorem)

Let

$\displaystyle \overline{X}_n = (X_{n1},\dots,X_{nk}) \;$ and $\displaystyle \; \overline{X} = (X_1,\dots,X_k)$

be k-dimensional random vectors. Then $\overline{X}_n$ converges to $\overline{X}$ in distribution if and only if

$\displaystyle \sum_{i=1}^k t_iX_{ni} \xrightarrow[n\rightarrow\infty]{D} \sum_{i=1}^k t_iX_i.$

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 $\overline{X}$ .

"Cramér-Wold theorem" is owned by Koro.

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Cross-references: distribution, coordinates, linear combination, converges, random vectors
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This is version 1 of Cramér-Wold theorem, born on 2002-12-10.
Object id is 3711, canonical name is CramerWoldTheorem.
Accessed 4490 times total.

Classification:

AMS MSC:

60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)

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