Let $(W_t)_{t\ge 0}$ be a Wiener process in $\mathbb R^d, d \ge 1$. For which $d$ holds $\lim_{t\to \infty} |W_t|=\infty$? Provide complete proof of this property/lack of this property for those $d$ for which it holds/does not hold.
I have no idea what properties of the Wiener process I could use to solve this problem. I tried using the law of the iterated logarithm, but didn't get anything helpful.