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?



Central limit theorem for negatively dependent random variables
Assuming $Y\_{2i+1} + Y\_{2i+2}$ is independent of $Y\_{2i+3} + Y\_{2i + 4}$, then the [answer to this question](https://mathoverflow.net/questions/29508/is-there-a-central-limit-theorem-for-bounded-non-identically-distributed-random) 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 $Y\_{2i}$.


