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A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.

Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and the central limit theorem? Is there a good reference for it? Thanks!

For the measure-theoretic entropy $h_{\mu}(T)$, see https://en.wikipedia.org/wiki/Measure-preserving_dynamical_system#Measure-theoretic_entropy

For the central limit theorem, means: for any smooth observable $\phi$ on $S^1$ with $\int \phi d\mu=0$, $\frac{\sum_{i \le n}\phi \circ T^i}{\sqrt{n}} \stackrel{d}{\longrightarrow} N(0,\sigma^2_{\phi})$ holds where $\sigma^2_{\phi} \ge 0$ is a constant.

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  • $\begingroup$ As I see it the normal distribution arises when minimizing the differential entropy subject to a given expectation and variance, when we don't know the expectation and variance what we minimize is the relative/cross entropy (KL divergence) and we get a normal distribution with 0 expectation estimator if unbiased and variance estimator given by the Fisher information... $\endgroup$
    – Dabed
    Commented May 5, 2020 at 9:51
  • $\begingroup$ ...then in CLT S converges to $N(\mu, \sigma)$ while the mle converges to $N(0,I(\mu,\sigma)^{-1})$ so I think some kind of similar play should be play between the measure entropy and the normal here $\endgroup$
    – Dabed
    Commented May 5, 2020 at 9:58
  • $\begingroup$ If the CLT as you stated it holds for $\phi$, it cannot hold for $2\phi$, right? $\endgroup$
    – Kostya_I
    Commented May 6, 2020 at 10:04
  • $\begingroup$ @Kostya I edited my question. $\endgroup$
    – jason
    Commented May 6, 2020 at 14:05

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