Exact rates in log law for positively associated random variables
Let \(\{X_{n}:n\geq 1\}\) be a strictly stationary sequence of positively associated random variables with \(EX_{1}=0\) and \(E\left| X_{1}\right| ^{2+\delta }<\infty \) for some \(\delta \in ]0,1],\) and set \( S_{n}=X_{1}+\dots+X_{n}\) and \(M_{n}=\max_{1\leq k\leq n}\left| S_{k}\right| .\) Assuming that \(\sum_{j\geq n}\text{Cov} (X_{1},X_{j})=O(n^{-\alpha })\) for some \(\alpha >1,\) the author provides precise asymptotics, as \(\varepsilon \searrow 0,\) of the series \(\sum_{n\geq 1}\frac{(\log n)^{b}}{n}P(\left| S_{n}\right| \geq \varepsilon \sigma \sqrt{n\log n})\) and \(\sum_{n\geq 1}\frac{(\log n)^{b}}{n}P(M_{n}\geq \varepsilon \sigma \sqrt{n\log n}),\) where \(b>-1\) and \(\sigma ^{2}=EX_{1}^{2}+2\sum_{j\geq 2}EX_{1}X_{j}<\infty ,\) as well as the a.s. and in \(L_{2}\) convergence, as \(\varepsilon \searrow 0,\) of the random series \( \sum_{n\geq 1}\frac{(\log n)^{b}}{n}I\{\left| S_{n}\right| \geq \varepsilon \sigma \sqrt{n\log n}\},\) where \(-1<b<0.\)
scientific article; zbMATH DE number 5563867