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).

  • $\begingroup$ This is far too much to ask in this level of generality. Definite counterexamples exist even for quite nice situations like an irrational rotation and $f$ a difference of characteristic functions of intervals. The kind of condition where theorems like this are known is: $T$ hyperbolic and $f$ smooth. $\endgroup$ – Anthony Quas Jan 18 '15 at 16:24
  • $\begingroup$ The sequence of random variables $(f\circ T^i)$ is strictly stationary, and there exists a vast litterature about the central limit theorem for stationary sequences. In this context, we can try (for example) a martingale approximation. In any case, I think you have to be more specific about the conditions you are looking for. $\endgroup$ – Davide Giraudo Jan 18 '15 at 16:25
  • $\begingroup$ @Anthony and Davide: I see; I wasn't aware of that. I have made a small edit; I am basically looking for some references on this subject. $\endgroup$ – Ritwik Jan 18 '15 at 16:38
  • $\begingroup$ @Anthony: I want to understand that last comment of yours: suppose X was the torus and T:X->X was the arnold cat map; are you saying there is a general theorem saying that the CLT for any smooth f would be satisfied? Here is the definition of arnold's cat map en.wikipedia.org/wiki/Arnold%27s_cat_map. $\endgroup$ – Ritwik Jan 18 '15 at 17:37
  • 3
    $\begingroup$ The key word if you want to look for yourself is Dynamical Central Limit Theorem. A survey (from 2005) by Melbourne and Nicol can be found at math.uh.edu/%7Enicol/psfiles/refineCLT.ps $\endgroup$ – Anthony Quas Jan 18 '15 at 18:00

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.