Donsker's theorem states that a random walk $X_n = \xi_1+...\xi_n$ with step size of mean 0 and variance 1, after rescaling by a factor of $\sqrt{n}$, converges to a Brownian motion weakly: $$ \left(\frac{X_{nt}}{\sqrt{n}}\right)_{t \geq 0} \overset{d}{\to} (B_t)_{t \geq 0}. $$
Is there an analog for Brownian motions in $\mathbb{R}^d$, $d \geq 2$?