Precise rates in log laws for NA sequences (Q2478369)

Summary: Let \(X_1,X_2,\dots\) be a strictly stationary sequence of negatively associated (NA) random variables with \(EX_1=0\), set \(S_n=X_1+\cdots+X_n\), suppose that \(\sigma^2= EX_1^2+2 \sum_{n=2}^\infty EX_1X_n>0\) and \(EX_1^2<\infty\), if \(-1<\alpha\leq1\); \(EX_1^2(\log|X_1|)^\alpha<\infty\), if \(\alpha>1\). We prove \(\lim_{\varepsilon\downarrow0} \varepsilon^{2\alpha+2} \sum_{n=1}^\infty ((\log n)^\alpha/n) P(|S_n|\geq \sigma(\varepsilon+\kappa_n) \sqrt{2n\log n})= 2^{-(\alpha+1)}(\alpha+1)^{-1} E|N|^{2\alpha+2}\), where \(\kappa_n= O(1/\log n)\) and \(N\) is the standard normal random variable.