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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 $A$.

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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.

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  • $\begingroup$ Thank you Did for pointing me in the right direction with this survey paper, this will save me much time in researching this question. I had posted my question with the hope that for the 'simple' case of bounded, uncorrelated RVs, there may be a well known result or theorem that someone more knowledgeable than me, like yourself, may have worked with before. From your comment on this being a venerable subject, I suppose it must also be a challenging subject that requires some endeavor even for seemingly 'simple' cases:) Thanks again for your advice, I appreciate it. $\endgroup$ – Hakeem May 16 '14 at 3:32

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