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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!
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!