On the convergence rate in the central limit theorem for associated processes (Q1111230)

Let \(X_ 1,X_ 2,..\). be associated random variables with E \(X_ j=0\), E \(| X_ j|^ 3\leq c<\infty\) and \(\sigma^ 2_ n=E S^ 2_ n\geq an\) with \(a>0\) and \(S_ n=X_ 1+...+X_ n\). Under the assumption of uniform exponential decrease of the covariance function of the X- process the author derives an estimate of Berry-Esséen type for the speed of convergence of \(P(S_ n\leq \sigma_ nx)\) to the standard normal distribution function. The order of this estimate is \(n^{- 1/2}(\log n)^ 2\). It is \(n^{-1/2}\log n\) if E \(| X_ j|^{3+\delta}\leq c\leq \infty\) for some \(\delta >0\). It is remarked that the only estimate known, given by \textit{T. E. Wood}, ibid. 11, 1042- 1047 (1983; Zbl 0522.60017), is of order \(n^{-1/5}.\) When the order of uniform decrease of the covariance function is an arbitrarily large negative power instead of exponential, there exists a process satisfying all other conditions where the convergence is slower than \(n^{-\rho}\) for some \(\rho\in (0,1/2)\). This is shown by construction of an example.