Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ (measure preserving means that $\mu(T^{-1}(A)) = \mu(A)$ for every measurable set $A$). Let $f: X \rightarrow \mathbb{R}$ be a measurable function that is in $L^1(X, \mu)$. Given an $x \in X$, define the following quantities $$ I := \int_X f d\mu, \qquad I_n(x) := \frac{\sum_{k=0}^{n-1} f(T^kx)}{n}. $$ By the ergodic theorem, we know that for almost all $x\in X$, $I_n(x)$ converges to $I$.

My question is now the following: under what hypothesis on $f$, can one claim that $$ \lim_{n \rightarrow \infty} \mu\{x \in X: a\leq \sqrt{n}(I-I_n(x)) \leq b)\} = \int_{a}^b G_{\sigma}(y) dy, $$ where $G_{\sigma}(y)$ is the Gaussian centered around $0$ with standard deviation $\sigma$? And moreover what will be that $\sigma$ (I would imagine there should be some formula for $\sigma$ in terms of $f$)?

To keep things simple, assume that $\int_X f^2 d\mu $ is finite (but ideally I would also like to know what is known about the rate of convergence when the integral of $f^2$ is not finite).

$\textbf{EDIT:} $ It has been pointed out that it is not realistic to expect an answer to this general question. I am therefore looking for references that address this question for specific examples of $X$ and $T$. Ideally, I am looking for a comprehensive survey article (that includes examples, counter examples and open questions) on this topic.

$\textbf{EDIT:}$ Examples involving $X:= [0,1]$ and $T$ being multiplication by some number modulo one are also fine (I had written earlier that I am not looking for that particular example; ignore that remark if you saw it).