If $\{S_n\}$ is a genuinely $d$-dimensional random walk and $d \geqq 3$, then with probability 1, $n^{-\alpha}|S_n| \rightarrow \infty$ as $n \rightarrow \infty$ for every $\alpha < \frac{1}{2}$. This follows from a recent result of H. Kesten. In this paper we show that, under certain conditions, there is a constant $\alpha_0$ depending on the walk, but $\frac{1}{2} - 1/d \leqq \alpha_0 < \frac{1}{2}$, and a deterministic sequence of vectors $\{\nu_n\}$ such that $\lim \inf_n n^{-\alpha}|S_n - \nu_n| = 0$ with probability 1 for every $\alpha \geqq \alpha_0$. In discrete time this phenomenon cannot occur for any $\alpha < \frac{1}{2} - 1/d$; in continuous time it can occur for any $\alpha > 0$.