# Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the sum, $\sum_{i=1}^N X_i$? In other words, is there a form of central limit theorem that applies to identical, non-independent (but uncorrelated) random variables that are bounded?

The best I've been able to find is the case of $M$-dependence for stationary RVs, where RVs more than $M$ apart (in time) can be assumed independent and thus CLT applies. I would greatly appreciate any suggestions or pointers!

Background:

Here is the background problem I am applying this to:

Given the arrays $C=[C_1,C_2,...,C_N]$ and $S=[S_1,S_2,...,S_N]$ of lengths $N$ with elements that are discrete iid uniform distributed with equal probability (p=1/2) of being $\pm$1.

Consider the sum below, for a constant $k \in [1,N]$:

\begin{equation*} A=\underset{m\neq l, n\neq l}{\underset{-l+m+n=k}{\sum_{l=1}^N \sum_{m=1}^N \sum_{n=1}^N}} C_lC_mC_n+S_lS_mC_n-C_lS_mS_n+S_lC_mS_n \end{equation*}

Each combination of $l, m, n$ results in a sum of four pairwise independent Bernoulli RVs, I denote this sum of four RVs as $X_i$ where $X_i=C_lC_mC_n+S_lS_mC_n-C_lS_mS_n+S_lC_mS_n$. The $X_i$ are thus identical, uncorrelated and bounded, $|X_i| \leq 4$.

The other option I am considering is breaking $A$ into two parts but I am still working on it:

\begin{equation*} A=Y+Z=\underset{m\neq l, n\neq l}{\underset{-l+m+n=k}{\sum_{l=1}^N \sum_{m=1}^N \sum_{n=1}^N}} C_lC_mC_n+ \underset{m\neq l, n\neq l}{\underset{-l+m+n=k}{\sum_{l=1}^N \sum_{m=1}^N \sum_{n=1}^N}} (S_lS_mC_n-C_lS_mS_n+S_lC_mS_n) \end{equation*}

Simulation results show good match to a Normal distribution for $A$.