Convergence rates in the complete moment of moving-average processes

In this paper, we discuss precise asymptotics for a new kind of moment convergence of the moving-average process $$X_k = \sum\limits_{i = - \infty }^\infty {a_{i + k} \varepsilon _i }$$, k ≥1, where {ε i: −∞ < i < ∞} is a doubly infinite sequence of independent identically distributed random variables with mean zero and the finiteness of variance, {α i: −∞ < i < ∞} is an absolutely summable sequence of real numbers, i.e., $$\sum\limits_{i = - \infty }^\infty {\left| {a_i } \right| < \infty }$$.