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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?

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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.

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