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# Cramér-Wold theorem

Let

$\overline{X}_{n}=(X_{{n1}},\dots,X_{{nk}})\;\mbox{and}\;\overline{X}=(X_{1},% \dots,X_{k})$ |

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

$\sum_{{i=1}}^{k}t_{i}X_{{ni}}\xrightarrow[n\rightarrow\infty]{D}\sum_{{i=1}}^{% k}t_{i}X_{i}.$ |

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 $\overline{X}$.

Type of Math Object:

Theorem

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Reference

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