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 $A$.
Best Answer
Central limit theorems under weak dependence conditions are a venerable subject with a long history. For a recent survey with probably more information than you care to know about, see Bradley, Richard (2005), Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions, Probability Surveys 2: 107–144. For a much lighter introduction and some references, start with the obvious.