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A Berry-Esseen theorem for weakly negatively dependent random variables and its applications. (English) Zbl 1121.60024
A pair of real random variables $$(X,Y)$$ is called negatively quadrant dependent (abbreviated: NQD) if $$\text{Cov}\,(f(X), f(Y))\leq 0$$ for all non-increasing functions $$f$$, $$g$$. A sequence $$(X_n)_{n\geq 1}$$ of random variables is called linearly NQD (LNQD), if for all disjoint set of integers $$A$$, $$B$$ and all $$r_i\geq 0$$ the random variables $$(\sum_{i\in A} r_i\cdot X_i, \sum_{i\in B} r_i\cdot X_i)$$ are NQD, cf. E. L. Lehmann [Ann. Math. Stat. 37, 1137–1153 (1966; Zbl 0146.40601)], C. M. Newman [Asymptotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability, Inst. Math. Stat. Hayward CA, pp. 127-140 (1984)].
For NQD sequences a version of the central limit theorem was proved by the second named author in [Acta Math. Hung. 86, 237–259 (2000; Zbl 0964.60035)]. In the paper under review the authors prove Berry-Esseén type estimates for the rate of convergence (Theorem 2.2 and 2.3).

##### MSC:
 60F05 Central limit and other weak theorems 60G50 Sums of independent random variables; random walks 60F17 Functional limit theorems; invariance principles 60E15 Inequalities; stochastic orderings
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