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

Suppose that the moving average process is based on a doubly infinite sequence of identically distributed and dependent random variables with zero mean and finite variance and that the sequence of coefficients is absolutely summable. Under suitable conditions of dependence, we show the precise rates in the law of logarithm of a kind of weighted infinite series for the first moment of the partial sums of the moving average process. This generalizes the common law of logarithm with the square root of logarithm to that with any positive power of logarithm. Moreover, we provide another law of logarithm as a supplement.