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

Authors:

Institution:

(1) Zhejiang University of Finance and Economics, Hangzhou 310012, P. R. China;(2) Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China

Abstract:

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. Research supported by National Natural Science Foundation of China

Keywords:

" target="_blank"> Moving-average process φ -mixing Negative association The law of the iterated logarithm

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