Moment convergence rates in the law of iterated logarithm for moving average process under dependence

Abstract

Abstract We assume that X k = ∑ i = − ∞ + ∞ a i ξ i + k $X_{k}=\sum_{i=-\infty}^{+\infty}a_{i}\xi_{i+k}$ is a moving average process and { ξ i , − ∞ < i < + ∞ } $\{\xi_{i},-\infty< i<+\infty\}$ is a doubly infinite sequence of identically distributed and dependent random variables with zero mean and finite variance and { a i , − ∞ < i < + ∞ } $\{a_{i},-\infty < i<+\infty\}$ is an absolutely summable sequence of real numbers. Under suitable conditions of dependence, we get the precise rates in the law of iterated logarithm for the first moment of the partial sums of the moving average process.

Keywords

• precise asymptotics
• the law of iterated logarithm
• moment convergence
• moving average
• φ-mixing