# Donsker's Theorem in Higher Dimensions

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 $\mathbb{R}^d$, $d \geq 2$?

• Do you mean a Brownian motion indexed by $\mathbb R^d$ or with values in $\mathbb R^d$? – Davide Giraudo Jun 27 '18 at 16:42
• $\mathbb{R}^d$-valued Brownian motion indexed by $t \in [0,\infty)$. – polar bear in a snowstorm Jun 27 '18 at 19:59
• If $X_n$ and $Y_n$ are independent random walks with step size of mean $0$ and variance $1$, then $\left( \frac{X_{nt}}{\sqrt{n}},\frac{Y_{nt}}{\sqrt{n}}\right)$ converges to a 2D Brownian motion in distribution. Is this right? – polar bear in a snowstorm Jul 2 '18 at 10:54