## Limiting behaviour of moving average processes under $$\varphi$$-mixing assumption.(English)Zbl 1154.60026

Summary: Let $$\{Y_i, - \infty <i<\infty \}$$ be a doubly infinite sequence of identically distributed $$\varphi$$-mixing random variables, $$\{a_i, - \infty <i<\infty \}$$ be an absolutely summable sequence of real numbers. In this paper we prove the complete convergence and Marcinkiewicz-Zygmund strong law of large numbers for the partial sums of moving average processes $$\{X_n = \sum_{i=-\infty}^\infty a_iY_{i+n}, n \geq 1\}$$ based on the sequence $$\{Y_i, - \infty <i<\infty \}$$ of $$\varphi$$-mixing random variables, improving the result of [Zhang, L., 1996. Complete convergence of moving average processes under dependence assumptions. Statist. Probab. Lett. 30, No. 2, 165–170 (1996; Zbl 0873.60019)].

### MSC:

 60F15 Strong limit theorems

Zbl 0873.60019
Full Text:

### References:

 [1] Burton, R.M.; Dehling, H., Large deviations for some weakly dependent random processes, Statist. probab. lett., 9, 397-401, (1990) · Zbl 0699.60016 [2] Chen, P.; Hu, T.-C.; Volodin, A., A note on the rate of complete convergence for maximums of partial sums for moving average processes in randemacher type Banach spaces, Lobachevskii J. math., 21, 45-55, (2006) · Zbl 1112.60023 [3] Hsu, P.L.; Robbins, H., Complete convergence and the law of large numbers, Proc. natl. acad. sci. USA, 33, 25-31, (1947) · Zbl 0030.20101 [4] Ibragimov, I.A., Some limit theorem for stationary processes, Theory probab. appl., 7, 349-382, (1962) · Zbl 0119.14204 [5] Li, D.; Rao, M.B.; Wang, X.C., Complete convergence of moving average processes, Statist. probab. lett., 14, 111-114, (1992) · Zbl 0756.60031 [6] Seneta, E., () [7] Shao, Q.M., A moment inequality and its application, Acta math. sinica, 31, 736-747, (1988), (in Chinese) · Zbl 0698.60025 [8] Zhang, L., Complete convergence of moving average processes under dependence assumptions, Statist. probab. lett., 30, 165-170, (1996) · Zbl 0873.60019
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