There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:

has a metric space $\Omega$, a map (or flow, but let's stick to maps) $T:\Omega\to\Omega$ and a (Borel) invariant probability measure $\mu$,

and aims at finding a suitable space $\mathcal{H}$ of functions $f:\Omega\to \mathbb{R}$ such that for all $f\in \mathcal{H}$, denoting by $X$ a random variable of distribution $\mu$, the sequence of random variables $f(X), f(T X), f(T^2 X),\dots$ satisfies the conclusion of the CLT for some variance $\sigma^2$.

I can see the beauty of this, first as it yields for a sequence of completely dependent random variables the conclusion of a Theorem most usually proved with a high degree of independence (thus enforcing the principle that deterministic chaos is tamed by time averaging); and second as it gives a strengthening of ergodicity, giving insight on the rate of equidistribution of orbits.

My question is whether there are application of this kind of CLT? By application I don't (necessarily) mean real-life or applied science application, but rather mathematical statements who do not look like they involve the CLT, but whose proof use it in an essential way.