Suppose we have some i.i.d. random variables $S_i$, $i\in\mathbb{N}$, $S_i \geq 0$ almost surely and we define $T_n := \sum_{i=1}^n S_i$ for $n \in \mathbb{N}$. Let $\epsilon > 0$ be arbitrary. What can we say about $\mathbb{P}(\exists k\in\mathbb{N}: t_k \in [t, t+\epsilon])$ as $t\to \infty$? My intuition says that $\lim_{t\to\infty} \mathbb{P}(\exists k\in\mathbb{N}: T_k \in [t, t+\epsilon]) = \epsilon$ if the $S_i$ are 'not structured enough'. For example, if $\mathbb{P}(S_i = 1) = 1$, then if $\epsilon$ is small enough, $\mathbb{P}(\exists k \in \mathbb{N}: T_k \in [t, t+\epsilon]) \in\{0,1\}$, depending on whether $\mathbb{N}\cap[t, t+\epsilon]$ is empty or not. However, if you take $S_i = U[0,1]$, then it feels as if it should not really matter which (large enough) $t$ you choose, the probability should only depend on the length of the interval you are considering.
Is there any rigorous treatment of this question? References to where I can find that are kindly appreciated. I have a feeling this is a typical entropy related question, however I am not familiar with good texts treating that topic.