Cramér-Wold theorem

Let

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

be random vectors. Then ${\overline{X}}_n$ converges to $\overline{X}$ in distribution (http://planetmath.org/ConvergenceInDistribution) if and only if

$$\sum _{i=1}^k{t}_i{X}_{ni}\underset{n\to \mathrm{\infty }}{\overset{𝐷}{\to }}\sum _{i=1}^k{t}_i{X}_i.$$

for each $(t_1,\mathrm{\dots },t_k)\in {ℝ}^k$. That is, if every linear combination of the coordinates of ${\overline{X}}_n$ converges in distribution to the correspondent linear combination of coordinates of $\overline{X}$.