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A central limit theorem for generalized quadratic forms. (English) Zbl 0596.60022
Random variables of the form $$W(n)=\sum_{1\leq i\leq n}\sum_{1\leq j\leq n}w_{ijn}(X_ i,X_ j)$$ are considered with $$X_ i$$ independent (not necessarily identically distributed), and $$w_{ijn}(.,.)$$ Borel functions, such that $$w_{ijn}(X_ i,X_ j)$$ is square integrable and has vanishing conditional expectations: $E(w_{ijn}(X_ i,X_ j)| X_ i)=E(w_{ijn}(X_ i,X_ j)| X_ j)=0,\quad a.s.$ A central limit theorem is proved under the condition that the normed fourth moment tends to 3. Under some restrictions the condition is also necessary. Finally conditions on the individual tails of $$w_{ijn}(X_ i,X_ j)$$ and an eigenvalue condition are given that ensure asymptotic normality of W(n).

##### MSC:
 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics
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