×

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

Citations:

Zbl 0873.60019
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.