1
$\begingroup$

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?

$\endgroup$

closed as off-topic by Douglas Zare, Marco Golla, Stefan Kohl, Yemon Choi, Myshkin Dec 24 '16 at 18:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Douglas Zare, Myshkin
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ 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
0
$\begingroup$

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 $Y_{2i}$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.