Some exact limit theorems for negatively associated sequences (Q2846875)

Let \(\alpha ,\delta >0\) and \(L=\delta ^{-1}\operatorname{E}\left| N\right|^{\alpha \delta +2},\) where \(N\) stands for a standard normal random variable. Let also \(w_{n}=(\log \log n)^{\delta -1}/n^{2}\log n,\) \(n\geq 3\). Let \(X_{1},X_{2},\dots\) be a stationary sequence of negatively associated random variables with \(\operatorname{E}X_{1}=0\), \(\operatorname{E}[X_{1}^{2}(\log \log (3+\left| X_{1}^{2}\right| ))^{\delta }]<\infty \) and \(0<\sigma^{2}=\operatorname{E}X_{1}^{2}+2\sum_{n\geq 2}\operatorname{E}X_{1}X_{n}<\infty\). Set \(S_{n}=X_{1}+\dots+X_{n}\) and \(M_{n}=\max_{1\leq k\leq n}\left| S_{k}\right|\). The author asserts that \(\lim_{\varepsilon \searrow 0}\varepsilon ^{\alpha \delta }\sum_{n\geq 3}w_{n}\operatorname{E}[S_{n}^{2}I\{\left| S_{n}\right| \geq \varepsilon \sigma \sqrt{n}(\log \log n)^{1/\alpha }\}]=\sigma ^{2}L\) and \(\lim_{\varepsilon \searrow 0}\varepsilon ^{\alpha \delta }\sum_{n\geq 3}w_{n}\operatorname{E}[M_{n}^{2}I\{M_{n}\geq \varepsilon \sigma \sqrt{n}(\log \log n)^{1/\alpha }\}]=2\sigma ^{2}L\sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)^{^{\alpha \delta +2}}}.\) Curiously enough, at some point, he erroneously considers a sufficient condition as being a necessary condition, and vice versa.