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I have $N$ Bernoulli random variables that each one of them is negatively dependent to exactly one of the other variables, for example: $Y_1$ is dependent to $Y_2$ and $Y_3$ is dependent to $Y_4$ and so on. All $Y_i$ with odd indexes are independent and identically distributed and all $Y_i$ with even indexes are independent and identically distributed, but distribution of the random variables with odd indexes is different than the random variables with even indexes.

Now my question is this: is central limit theorem true for these variables?

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    $\begingroup$ So what happens if $Y_1, Y_3, Y_5, \dots$ are iid with some (non-symmetric) distribution, and $Y_2 = -Y_1$, $Y_4 = -Y_3$, etc? Then we definitely will not get the CLT... $\endgroup$ – Nate Eldredge Dec 23 '16 at 4:49
  • $\begingroup$ I'm voting to close this question because it does not seem to have been thought through sufficiently $\endgroup$ – Yemon Choi Dec 23 '16 at 18:07
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Assuming $Y_{2i+1} + Y_{2i+2}$ is independent of $Y_{2i+3} + Y_{2i + 4}$, then the answer to this question answers yours in the affirmative. To prevent non-degeneracy, your condition that "distribution of the random variables with odd indexes is different than the random variables with even indexes" is necessary, so that $\sigma(\sum_i Y_i) \to \infty$.

Update: as Nate pointed out in the comment, you also need that $-Y_{2i+1}$ has different distribution from $Y_{2i}$.

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