Suppose there is a triangular array of random variables $X_{k,n}$ with the following properties: $$ \mathbb{P}(X_{k,n} = 1) = p_n, \quad \mathbb{P}(X_{k,n} = 0) = 1 - p_n, \quad \lim_{n\to\infty}p_n = p > 0,\\ \mathrm{cov}(X_{i,n}, X_{j,n}) = \begin{cases} c_n \neq 0, & |i - j| \leq 1, \\ d_n, & |i - j| > 1. \end{cases}, \quad \lim_{n\to\infty}\mathrm{cov}(X_{i,n}, X_{j,n}) = \begin{cases} c \neq 0, & |i - j| \leq 1, \\ 0, & |i - j| > 1. \end{cases} $$ To be exact, $d_n$ decays as $O(n^{-1})$.
Let's define $S_n = \sum_{k=1}^n X_{k,n}$, $\mu_n = \mathbb{E}S_n$ and $\sigma_n^2 = \mathrm{var}S_n$. It's also known that $\mu_n$ and $\sigma_n^2$ are both $O(n)$. From numerical simulations, I can see that $\frac{S_n - \mu_n}{\sigma_n}$ converges in distribution to $\mathcal{N}(0,1)$.
I'm looking for a reference to some version of Central Limit Theorem for this setup. I'm not sure if given properties are enough to prove the convergence, so having a theorem that requires additional facts about dependence is fine as well.