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

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.