# Cramér-Wold theorem

Let

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

be random vectors. Then $\overline{X}_{n}$ converges to $\overline{X}$ in distribution (http://planetmath.org/ConvergenceInDistribution) 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 linear combination of the coordinates of $\overline{X}_{n}$ converges in distribution to the correspondent linear combination of coordinates of $\overline{X}$.

Title Cramér-Wold theorem CramerWoldTheorem 2013-03-22 13:14:21 2013-03-22 13:14:21 Koro (127) Koro (127) 4 Koro (127) Theorem msc 60E05