# Converse for the central limit theorem of $q$-dependent random variables

Let $$(X_n)_n$$ be a sequence of $$q$$-dependent random variables and identically distributed. If $$E[X_1^2]<+\infty,$$ then the Hoeffding-Robbins theorem states that $$\frac{1}{\sqrt{n}}\sum_{k=1}^n(X_k-E[X_k]) \Rightarrow N(0,\sigma^2).$$

The converse seems to be true, that is if $$\frac{1}{\sqrt{n}}\sum_{k=1}^nX_k$$ converges in distribution to a random variable $$X$$ then $$E[X_1^2]<+\infty,$$ in particular when $$q=0$$: https://math.stackexchange.com/questions/3202451/central-limit-theorem-and-integrability/3623412#3623412 and https://math.stackexchange.com/questions/3961865/central-limit-theorem-for-weighted-random-variable

The converse, does it hold if $$q>0$$ ? Is there any proof, paper, reference... for this?

Probably not, there are trivial case like $$X_1, -X_1, X_2, -X_2,...$$ with the X's symmetric to make them identically distributed, but even if you have some good way to exclude them there are probably subtler cases where only the q-fold sums are at all well behaved.