**Motivation:**
If I recall correctly, the simple symmetric random walk from i.i.d binary steps converges in distribution to the Wiener measure (if scaled with $a_n = \sqrt{n}$). What I am wondering is if something like that can be true for the simplest opposite example I can think of: walks constructed from irrational rotation.

Let $\alpha$ be an irrational number and $T_\alpha: S^1 \to S^1$ be the rotation by $\alpha$, i.e., the transformation $T_\alpha$ is defined by $$z = e^{2\pi i \cdot \theta} \implies T_\alpha(z) = e^{2\pi i \cdot (\theta + \alpha)}$$.

Let $f: S^1 \to \{1,-1\}$ be the measurable function defined by assigning +1 to the first half $\{\exp(2\pi i \cdot \theta): \theta \in [0, \frac12]\}$ and -1 to the remaining half. We consider $\alpha$ to be a fixed parameter. The triple $(S^1, T_\alpha, f)$ defines a binary-valued stationary process:

$$z \in S^1 \mapsto (f(z), f(T(z)), f(T^2(z)) \dots) \in \{+1,-1\}^{\mathbb N}$$

where the ambient probability space for the process in our construction is $S^1$ equipped with the uniform probability distribution on it (its Haar measure).

Consider the random walk constructed from partial sums of this stationary process, that is, we define $$S_n(z) := f(z) + f(Tz) + \dots + f(T^{n-1}z)$$ and consider the new process: $$z \in S^1 \mapsto (0, S_1(z),S_2(z),\dots) \in {\mathbb Z}^{\mathbb N}$$

For the purpose of the following question, we extend the discrete-time random walk $\{S_n\}_{n \in \mathbb N}$ to the continuous-time random walk $\{S_t\}_{t \in [0,\infty)}$ by extending in the zig zag way.

**Q1.** As $n \to \infty$, does this random walk approach some distribution (with appropriate scaling)? In other words, is there a sequence $0 < a_n \nearrow \infty$ (maybe depending on $\alpha$) such that the path-valued random variable $B_N: S^1 \to C[0,\infty)$ defined by
$$B_N(z)(t) = \frac{S_{tN}(z)}{a_N}$$
converges in distribution to some nontrivial probability measure on $C[0,\infty)$ (which may depend on $\alpha$) as $N\to\infty$?

**Q2.** If the answer to Q1 is no for some $\alpha$, what are some sufficient conditions on the parameter $\alpha$ to ensure existence of a limiting distribution?
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