CLT for identically distributed but NOT independent random variables

Question

The random variables are not IID; in fact they are identically distributed but NOT independent. Can you give me a reference for CLT under this type of set-up?

Answer 1

Little can be said about what happens here in general. For example, suppose $X_k=(-1)^{k-1}X_1$, where $X_1$ has any symmetric distribution. Then $\sum_{k=1}^nX_1$ is $0$ ($X_1$) when $n$ is even (odd).

Answer 2

If variables in the sequence that are far away from each other are sufficiently close to independent, roughly the same results hold.

For example, if the sequence is stationary with pairwise correlations $\rho_k$, the limit of the scaled sum is Gaussian with variance

$$ \sigma^2+2\sum_{n>1}\rho_n. $$

Obviously this can only be true if the sum above exists, i.e., if correlations decay sufficiently quickly.

Formalizations of this are called "Ergodic CLT" and you can pick an exposition at your desired level from a Google search.