Let $X_1, \ldots, X_N$ be a string random variables taking values $X_i \in [-1,1]$ and jointly distributed according $P(X_1, X_2, \ldots, X_{N-1}, X_N)$, which is invariant under cyclic permutations of the variables, that is $P(X_1, X_2, \ldots, X_{N-1}, X_N) = P(X_2, X_3, \ldots, X_N, X_1)$. For simplicity let $\langle X_i \rangle=0$. We assume that these random variables are weakly dependent in the sense that their correlation function decays exponentially $$\left| \langle X_1 X_2 \cdots X_r\, X_{l+1} X_{l+2} \cdots X_{l+r} \rangle - \langle X_1 X_2 \cdots X_r \rangle \langle X_{l+1} X_{l+2} \cdots X_{l+r} \rangle \right| \leq e^{-\min\{l-r,\, N-l-r\}/\xi}$$ where $\xi >0$ is the correlation length and $l>r>0$ and $r+l < N$ are integers. In particular, this implies the usual $\left|\langle X_i X_j \rangle \right| \leq e^{-|i-j|/\xi}$ from which it follows that the variance is proportional to $N$.
I am looking for a bound on ${\rm prob}\{X_1+\cdots + X_N \geq \mu \sqrt{N}\} \leq \epsilon (\mu, N)$ like the exponential Chebyshev or Hoeffding inequalities, but for the weakly dependent case described above. This has been shown in two sub-cases, namely, when the string is a Markov chain, and when the distribution $P$ has a matrix product structure.
