On moments of the maximum of partial sums of moving average processes under dependence assumptions

Let {Y _i ;−∞ < i < ∞} be a doubly infinite sequence of identically distributed φ-mixing random variables and let {a _i ;−∞ < i < ∞} be an absolutely summable sequence of real numbers. In this paper we study the moments of $\mathop {\sup }\limits_{n \geqslant 1} \left| {\sum\limits_{k = 1 - \infty }^n {\sum\limits_{}^\infty {a_i Y_{i + k} /n^{1/r} } } } \right|^p (1 \leqslant r < 2,p > 0)$\mathop {\sup }\limits_{n \geqslant 1} \left| {\sum\limits_{k = 1 - \infty }^n {\sum\limits_{}^\infty {a_i Y_{i + k} /n^{1/r} } } } \right|^p (1 \leqslant r < 2,p > 0) under the conditions of some moments.