TITLE: What time does it take for irrational rotations to hit an interval?
QUESTION [13 upvotes]: Hi,
Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take this process to hit $(0,\alpha)$ ? From the ergodic theorem I know that, if I denote $N(n)$ the number of times $\theta_n \in (0,\alpha)$, then $N(n)/n \to \alpha$. What I want to know is how much time it will take for this limit to be attained.
Another way of framing this question is : is there any "central limit theorem" (or weakening thereof ; I'm mainly interested in guaranteed bounds for $P(N\geq 1)$) for ergodic processes? From what I've read, there is no general answer to this for a generic ergodic process and function f. There are some results that depend on $f$ being smooth, which it isn't here.
The same question was asked on Quantitative versions of ergodic theorem, but I haven't found anything there that relates to my question.

REPLY [10 votes]: There is a theorem of Kesten, which roughly says, that if you take $(\theta, \theta_0)$ random, and the number of times you hit $(0, \alpha)$ in the first $N$ iterations, subtract the expected $N \times \alpha$, and normalize by $\rho \times ln(n)$, the result will converge to Cauchy distribution. This can be viewed as an analogue of CLT in this case.