Suppose $x < y \in [0, 1]$ and $U\_1, U\_2, ...$ are random variables distributed as uniform on $(0, w)$ with $w < 1$. Define $S\_n = \sum\_{i = 1} ^ n U\_i$. We'll say that $(S\_i)$ splits $x$ and $y$ if, for some $n$, $x < S\_n < y$. I'm interested in what sorts of things could be said about the probability that $x$ and $y$ are split. If it helps, take $w$ to be small and $x, y$ close to $1$ and $|x - y| < w$.

I feel that this is quite related to whatever subject studies things like Poisson processes. Obviously, I know that if $|x - y| > w$ then $x$ and $y$ will be split, but beyond that I'm not sure what theory I would look to, and I don't know whether this sort of problem is tractable or not (it might be trivial for all I know). Only guess that I have is that near $|x - y| = w$ the probability should go to $0$ like $1 - \frac{|x - y|}{w}$, and this should be a lower bound for any $|x - y| < w$. I may also be interested in results or pointers for $U\_i$ not uniform, but I need it to be the case that for points sufficiently far away the probability of being split is exactly $1$. 

**EDIT**: In light of the solution posted by PinkElephant below, I'll reword this as follows. Set $w = 1$ and remove the restriction on $x, y \in [0, 1]$ so that just $x < y \in \mathbb R$. What is the probability that $x$ and $y$ are split for large values of $x, y$ but for $|x - y|$ fixed, i.e. $x, y \to \infty$ but $y - x$ constant. It seems to me that asymptotically things should only depend on $y - x$; for $x, y$ near $0$ it seems as though there is dependence on the fact that we started the sum from $0$ that I think we escape from for large $x, y$. 

**Update**: I have a heuristic argument that gives the asymptotic probability of $x$ and $y$ being split as $1 - (1 - |x - y|)^2\_+$ where $(a)\_+$ denotes $\max\{0, a\}$. The argument is as follows: it should not matter asymptotically where $x$ and $y$ lie, so we can assume $x= k$ for some positive integer $k$. $x$ and $y$ will be split if, for the first value of $S\_i$ in the interval $[x, x + 1]$, we have $x \le S\_i \le y$. Consider the Markov chain $T\_j$ consisting of the fractional parts of those $S\_i$ such that $(S\_{i - 1}) > (S\_i)$, where $(a)$ denotes the fractional part of $a$. The probability that $x$ and $y$ are split should be $P(T \le (y - x))$ where $T$ is drawn from the stationary distribution of the $T\_j$. I took a blind guess that the stationary distribution was given by the density $f(t) = 2(1 - t)$, drew a bunch of samples from the Markov chain I described, and verified empirically that $f(t) = 2 (1 - t)$ is the answer I'm supposed to get. 

If anyone wants to take a stab at verifying that $1 - (1 - |x - y|)\_+^2$ is indeed the correct answer asymptotically, it would be much appreciated.