A Closer Look at The Fine Structure of The Random Walk on (0,1)

Let \( S_n \) be the random walk on \( (0, 1) \). The \( S_n \) have been the subject of intense study; their definition is immediately intuitive. Nevertheless, they are quite intentionally disorderly, and this disorder is mirrored by the fact that, pointwise, \( \left( \frac{S_n}{\sqrt{n}} \mid n \in \mathbb{N}^+ \right) \) behaves quite badly.

In this paper, we provide our results on the fine structure of the random walk that give insight into this behavior.