Suppose $X_1,...,X_n$ is a sequence of stationary correlated random variables in $\{-1,+1\}$ such that : $\mathbb{P}[X_i = +1] = p$, $\mathbb{P}[X_i = -1] = 1-p$ with $p\in (0,1)$, and with a correlation $\rho_{j}$ is defined by: $$ \rho_{j}:= \frac{\mathbb{E}[X_iX_{i+j}] - \mathbb{E}[X_i]\mathbb{E}[X_{i+j}]}{\mathbb{E}[X_i]\mathbb{E}[X_{i+j}]}\text{ (independent of }i\text{ by stationarity)}$$

My question concerns the expectation of the product: $$P_n:=\mathbb{E}\left[\prod_{i=1}^n X_i\right]$$

Clearly, if the $X_i$ are independent, then $P_n = (2p-1)^n$ .

My question is the following:

If $\rho_j =O(e^{-\kappa j})$ when $j\to\infty$ for some $\kappa>0$, what can we say about $P_n$? Can we compute it explicitly ? Can we obtain an asymptotic formula in the limit $n\to \infty$?

Thanks for your help !

**Edit:**

A more specific question (proof or counter-example):

Is it true that one can find a sequence $c_n = O(e^{-\lambda n})$ for some $\lambda>0$ such that $$P_n = \prod_{i=1}^n (2p-1+c_i)$$