Precise asymptotics of complete moment convergence on moving average (Q1941315)

Let \(\{a_{i},\;i\in\mathbb{Z}\}\) be a sequence of real numbers such that \(\sum_{i\in \mathbb{Z}}|a_{i}|<\infty\), let \(\{\xi_{i}, i\in Z\}\) be a sequence of identically distributed \(\varphi\)-mixing random variables with \(\operatorname{E}\xi _0=0\) and \(\operatorname{E}\xi_0{}^2<\infty\), and put \(X_{k}=\sum_{i\in \mathbb{Z}}a_{i}\xi_{i+k}\), \(k\geq 1\), and \(S_{n}=\sum_{k=1}^nX_{k}\), \(n\geq 1\). On account of hard analysis, the authors extend three theorems in the precise asymptotics area due to \textit{W. D. Liu} and \textit{Z. Y. Lin} [Statist. Probab. Lett. 16, 1787--1799 (2006; Zbl 1104.60015)] involving expectations of \(|S_{n}|^{p}\operatorname{1}_{\{|S_{n}|\geq n\varepsilon\}}\), \(0\leq p\leq 2\), and \(S_{n}^2\operatorname{1}_{\{|S_{n}|\geq \sqrt{(n\log n)\varepsilon}\}}\) as \(\varepsilon\searrow 0\). Liu and Lin obtained similar results in the case that \(S_{n},\;n\geq 1\), is a random walk with i.i.d. steps.