Convergence rate estimate for a class of random fields in the central limit theorem (Q1898564)

Let \(\{X_j,j \in Z^d\}\) be a field of zero-mean associated random variables [see \textit{J. D. Esary}, \textit{F. Proschan} and \textit{D. W. Walkup}, Ann. Math. Stat. 38, 1466-1474 (1967; Zbl 0183.21502)], each having finite variance such that for some \(s \in(2,3]\) and some \(\lambda > d\), \(M_s : = \sup_j {\mathcal E} |X_j |^s < \infty\); also \[ u(n) : = \sup_{j \in Z^d} \sum_{q \in Z^d,|q - j |\geq n} \text{cov} (X_j, X_q) = O(n^{-\lambda}) \quad \text{as} \quad n \to \infty. \] Then for an arbitrary finite set \(V \subset Z^d\), \(\sup_{x \in \mathbb{R}} |P(S (V) \leq x ({\mathcal E} (S(V))^2)^{1/2} - \Phi (x)) |\) is bounded above by \(a ({\mathcal E} (S(V))^2)^{- s/2} |V |^{1 + \beta (s - 1)}\) where \(a\) is independent of \(V\) (depending only on \(M_s, d,u (\cdot)\), \(s, \beta)\) with \(\beta > {1 \over 2} (1 - {1 \over 2db + 1})\) and \(b : = \max \{{s_1 \over \lambda (s - 2)}, {3s \over 2 (\lambda - d)}, {2s \over (s - 1) (\lambda - d)}\}\).