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Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of attraction?

As a related question, if $X_1, X_2, X_3$ are identically distributed and centered and $E(X_i X_j) = c$, what bound can I get for $E(X_1 X_2 X_3)$ in terms of $c$?

Say I have $X_1, X_2, X_{ij}$, $j \ldots$ le i$with the property that$X_i$X_{ij}$ are centered and identically distributed and $E(X_i X_jE(X_{ij} X_{ij'}) = o(\exp(-n))$o(\exp(-i)))$. Then does$\sum_i X_i$\sum_j X_{ij}$ have Gaussian domain of attraction?
Say I have $X_1, X_2, \ldots$ with the property that $X_i$ are centered and identically distributed and $E(X_i X_j) = o(\exp(-n))$. Then does $\sum_i X_i$ have Gaussian domain of attraction?