Central limit theorem for weak correlated random variables

Question

I have a sequence of weak correlated continuous random variables $\{X_i\}$ with bounded variance and $\operatorname{Cov}(X_i,X_j)\rightarrow0$ for $|i-j|\rightarrow\infty$.

I was able to find a reference for the law of large numbers that work for above type of weakly correlated random variables.

I am trying to understand if there is a version of the central limit theorem that works for this kind of weakly correlated random variables.

I searched online and in one forum, I found that if covariance decays CLT works. However, I am not able to find the correct reference or theorem.

[EDIt] Added that $X_i$'s are continuous random variables

Answer 1

Even if the sequence $(X_i)$ is stationary with finite moments of any order and weakly dependent in every reasonable sense (so that, in particular, the condition $Cov(X_i,X_j)\to0$ as $|i-j|\to\infty$ holds), this is still not enough for any central limit theorem to hold.

E.g., for all integers $i$, let $X_i:=Y_i-Y_{i-1}$, where the $Y_i$'s are independent Rademacher random variables (r.v.'s), so that $P(Y_i=\pm1)=1/2$ for all $i$. Then $$\sum_{i=1}^n X_i=Y_n-Y_0,$$ which equals $Y_1-Y_0$ in distribution. Here, instead of assuming that the $Y_i$'s are Rademacher r.v.'s, we may assume that the distribution of each $Y_i$ is any distribution (with finite moments of any order) that is not normal (and not degenerate), so that the distribution of $Y_1-Y_0$ be not normal (or degenerate).

This counterexample is given in the beginning of Ch. 18 of the book by Ibragimov and Linnik, where positive results are also given. See also some of the great many references to this book; MathSciNet counts over 600 such references.

In particular, see Theorem 1.3 and Corollary 1.1 on p. 1339, stating a central limit theorem assuming certain moment and mixing conditions and, importantly, that $\liminf_n (Var\,\sum_1^n X_i)/n>0$.