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](https://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem), but I haven't found anything there that relates to my question.



What time does it take for irrational rotations to hit an interval?
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. 


