Limiting dist for sum of dependent and non-identical Bernoulli vars

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The discussion centers on the limiting distribution of the sum of Bernoulli variables, particularly in cases where the variables are independent but non-identically distributed (Poisson-Binomial distribution) and where they are dependent and non-identically distributed. For independent non-identical Bernoulli variables, the Lindeberg-Feller Central Limit Theorem applies, suggesting that the normalized sum converges to a standard normal distribution. However, the case of dependent non-identical Bernoulli variables remains uncertain, with a request for references or conditions under which a Central Limit Theorem might apply. Participants suggest exploring literature by Robert J. Serfling and Kai Lai Chung for insights on CLTs related to dependent summands. The need for clarification on the conditions for case ii) is emphasized.
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A Binomial distribution has a standard normal limiting distribution, i.e. (X-E[X])/se(X) -> N(0,1), where X is the sum of independent and identically distributed Bernoulli variables.

Does this hold even when
i) the Bernoulli variables are independent but non-identically distributed? That is, say that each Bernoulli variable have different survivor intensity and define X as the sum of these non-identical variables. I believe this distribution is called the Poisson-Binomial distribution. Do we have: (X-E[X])/se(X) -> N(0,1)?

ii) the Bernoulli variables are dependent and non-identically distributed? That is, say that each Bernoulli variable have different survivor intensity and that they are correlated. Define X as the sum of these non-identical variables. Do we have: (X-E[X])/se(X) -> N(0,1)?

The case ii) is what I'm mainly interested in. I'm pretty sure case i) holds, but isn't 100% case ii) holds. If it holds I would appreciate a reference to any paper or so since I need the conditions under which it holds.

Thanks!
 
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I would suggest, at least for (i), looking to see whether the conditions of the Lindeburg-Feller Central Limit Theorem are satisfied (I'm guessing they are but haven't worked through it). You can find an excellent discussion of this, and related theorems, in Robert J. Serfling's 'Approximation Theorems of Mathematical Statistics'. In my edition the discussions are on pages 28 through 32.

You could also look in Kai Lai Chung's `A Course In Probability Theory', which has a more extensive discussion of CLTs, including a section on dependent summands.
 
Yes, the Lindeberg-Feller CLT works fine for (i).

Does anyone have any idea for case (ii)? That is, is there any CLT for the case the bernoulli trials are dependent with different success probabilities.
 
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