Let

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

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 $\overline{X}$.