# Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

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## Abstract

In this paper, we discuss the moving-average process $$X_{k} = {\sum\nolimits_{i = - \infty }^\infty {\alpha _{{i + k}} \varepsilon _{i} } }$$, where {α i ;-∞ < i < ∞} is a doubly infinite sequence of identically distributed φ-mixing or negatively associated random variables with mean zeros and finite variances, {α i ;-∞ < i < ∞} is an absolutely summable sequence of real numbers. Set $$S_{n} = {\sum\nolimits_{k = 1}^n {X_{k} ,n \geqslant 1} }$$. Suppose that $$\sigma ^{2} = E\varepsilon ^{2}_{1} + 2{\sum\nolimits_{k = 2}^\infty {E\varepsilon _{1} \varepsilon _{k} } } > 0$$. We prove that for any $$\delta \geqslant 0,\;{\text{if}}\;E{\left[ {\varepsilon ^{2}_{1} {\left( {\log \;\log {\left| {\varepsilon _{1} } \right|}} \right)}^{{\delta - 1}} } \right]} < \infty$$,

$${\mathop {\lim }\limits_{ \in \searrow o} } \in ^{{2\delta + 2}} {\sum\limits_{n = 1}^\infty {\frac{{{\left( {\log \;\log \;n} \right)}^{\delta } }} {{n\;\log \;n}}} }P{\left\{ {{\left| {S_{n} } \right|} \geqslant \varepsilon \tau {\sqrt {2n\;\log \;\log \;n} }} \right\}} = \frac{1} {{{\left( {\delta + 1} \right)}{\sqrt \pi }}}\Gamma {\left( {\delta + 3/2} \right)},$$

, and if $$E{\left[ {\varepsilon ^{2}_{1} {\left( {\log {\left| {\varepsilon _{1} } \right|}} \right)}^{{\delta - 1}} } \right]} < \infty$$,

$${\mathop {\lim }\limits_{ \in \searrow o} } \in ^{{2\delta + 2}} {\sum\limits_{n = 1}^\infty {\frac{{{\left( {\log \;n} \right)}\delta }} {n}} }P{\left\{ {{\left| {S_{n} } \right|} \geqslant \varepsilon \tau {\sqrt {n\;\log \;n} }} \right\}} = \frac{{\mu ^{{{\left( {2\delta + 2} \right)}}} }} {{\delta + 1}}\tau ^{{2\delta + 2}} ,$$

where $$\tau = \sigma \cdot {\sum\nolimits_{i = - \infty }^\infty {\alpha _{i} ,\Gamma {\left( \cdot \right)}} }$$ is a Gamma function and μ(2δ+2) stands for the (2δ + 2)-th absolute moment of the standard normal distribution.

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Correspondence to Yun Xia Li.

Research supported by National Natural Science Foundation of China

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Li, Y.X., Zhang, L.X. Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes. Acta Math Sinica 22, 143–156 (2006). https://doi.org/10.1007/s10114-005-0542-4

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• DOI: https://doi.org/10.1007/s10114-005-0542-4