The Davis-Gut law for moving average processes
Let \(\{Y_i: -\infty<i<\infty\}\) be a sequence of independent and identically distributed random variables. Let \(\{a_i: -\infty<i<\infty \}\) be a sequence of real constants such that \(\sum_{i=-\infty}^{\infty}|a_i|<\infty\), and \(a=\sum_{i=-\infty}^{\infty}a_i \neq 0\). Now define \(X_n=\sum_{i=-\infty}^{\infty} a_{i+n} Y_i\), which is called the moving average process of \(\{Y_i: -\infty<i<\infty \}\). The authors prove the following main result. Theorem. Let \(h(\cdot)\) be a positive nondecreasing function on \((0,\infty)\) such that \(\int_{1}^{\infty} (t h(t))^{-1} dt = \infty \). Let \(\Psi(t)=\int_{1}^{t} (s\, h(s))^{-1} \,ds\). For \(\{X_n: n\geq 1 \}\) defined as above we have \[ \sum_{n=1}^{\infty} \frac{1}{nh(n)}\operatorname{P}\Bigg\{\bigg |\sum_{i=1}^{n}X_i \bigg |>(1+\epsilon) \sqrt{2n \log \Psi(n)}\Bigg\}\begin{cases} <\infty &\text{ if } \epsilon >0, \\ =\infty &\text{ if }\epsilon <0,\end{cases} \] if and only if \(\operatorname{E}Y_1=0 \text{ and } \operatorname{E}Y_1^2=1\). A couple of corollaries are also obtained when \(h(n)\) takes special forms such as (i) \(h(n)=(\log\log n)^{b}\), \(b\geq 0\), (ii) \(h(n)=(\log n)^{r}\), \(0\leq r<1\), and \(h(n)=\log n\).
scientific article; zbMATH DE number 6473533
scientific article