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