Complete convergence of moving average processes. (English) Zbl 0756.60031

Li, Deli; Bhaskara Rao, M.; Wang, Xiangchen

Stat. Probab. Lett. 14, No. 2, 111-114 (1992).

\([Y_ i\); \(-\infty<i<\infty]\) is a doubly infinite sequence of i.i.d. random variables; \([a_ i\); \(-\infty<i<\infty]\) is an absolutely summable sequence of real numbers, and \(X_ k\) is defined as \(\sum_{i=-\infty}^ \infty a_{i+k}Y_ i\), for \(k\geq 1\). Suppose \(r\), \(t\) are values satisfying \(1\leq t<2\) and \(r>1\). Then it is shown that \(E(Y_ 1)=0\) and \(E| Y_ 1|^{rt}<\infty\) imply \[ \sum_{n\geq 1}n^{r-2}P\left[\left|\sum_{k=1}^ n X_ k\right|>\varepsilon n^{1/t}\right]<\infty\qquad\text{for all }\varepsilon>0. \]

Reviewer: L.Weiss (Ithaca)

MSC:

60F15	Strong limit theorems
60G50	Sums of independent random variables; random walks

Keywords:

complete convergence; moving average processes; doubly infinite sequence of i.i.d. random variables

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