Suppose $\epsilon_1,\epsilon_2,...$ are i.i.d bounded random variables with compact support. Let $X_k=g_k(\epsilon_k,...,\epsilon_1)$ be Bernoulli random variables with the covariance between $X_i$ and $X_j$ decaying exponentially.

Does a CLT hold here for the $X_k$, i.e. does $\frac{X_1+...+X_n-\Sigma_{i=1}^n \mathbb{E}[X_i]}{\sqrt{n}}$ converge in distribution to $N(0,\sigma^2)$?

Is there any paper that adresses the subject?