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

1. May 21, 2013

### PAHV

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!

2. May 21, 2013

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.

3. May 22, 2013

### PAHV

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.