Complete convergence of moving average processes under dependence assumptions. (English) Zbl 0873.60019

Zhang, Li-Xin

Stat. Probab. Lett. 30, No. 2, 165-170 (1996).

The author considers a moving-average process \(X_k=\sum_{i=-\infty}^\infty a_{i+k}Y_i\), \(k\geq 1\), for an absolutely summable sequence \(\{a_i\}\) of real numbers and a doubly infinite sequence \(\{Y_i\}\) of identically distributed and \(\varphi\)-mixing random variables. Under a slight condition on the \(\varphi\)-mixing coefficient and \(EY_1=0\), \(E|Y_1|^{rt}<\infty\) for \(1\leq t<2\), \(r\geq 1\), the convergence of the series \(\sum_{n=1}^\infty n^{r-2}P(|\sum_{k=1}^n X_k|\geq n^{1/t}\varepsilon)\) for all \(\varepsilon>0\) could be proved.

Reviewer: L.Heinrich (Freiberg)

MSC:

60F15	Strong limit theorems
60G50	Sums of independent random variables; random walks
60G05	Foundations of stochastic processes

Keywords:

complete convergence; phi-mixing; moving-average process

References:

[1]	Burton, R. M.; Dehling, H., Large deviations for some weakly dependent random process, Statist. Probab. Lett., 9, 397-401 (1990) · Zbl 0699.60016
[2]	Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, (Proc. Nat. Acad. Sci., 33 (1947)), 25-31 · Zbl 0030.20101
[3]	Ibragimov, I. A., Some limit theorems for stationary processes, Theory Probab. Appl., 7, 349-382 (1962) · Zbl 0119.14204
[4]	Li, D. L.; Rao, M. B.; Wang, X. C., Complete convergence of moving average processes, Statist. Probab. Lett., 14, 111-114 (1992) · Zbl 0756.60031
[5]	Shao, Q. M., A moment inequality and its application, Acta Math. Sinica, 31, 736-747 (1988), (in Chinese) · Zbl 0698.60025
[6]	Shao, Q. M., Almost sure invariance principles for mixing sequences of random variables, Stochastic Processes Appl., 48, 319-334 (1993) · Zbl 0793.60038