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If $X_n\Rightarrow X$, the continuous mapping theorem givesfor every $u$. Conversely, the assumed one-dimensional convergence at $t=1$ givesfor every $u\in {\mathbb{R}}^d$. Lévy's convergence theorem then yields $X_n\Rightarrow X$. This is the Cramer-Wold theorem.

For real $u_1,\dots ,u_j$, the preceding series construction givesIt is centered Gaussian, and Parseval identity makes its varianceThis is the variance of ${\sum }_r{u}_r{B}_{t_r}$. The Cramer-Wold theorem proves equality of the two joint distributions.