Previous |  Up |  Next

Article

Keywords:
moving average; $\varphi $-mixing; complete convergence; $q$-order moment; maximum of partial sums
Summary:
Let $\{Y_i, -\infty <i<\infty \}$ be a doubly infinite sequence of identically distributed $\varphi $-mixing random variables, and $\{a_i, -\infty <i<\infty \}$ an absolutely summable sequence of real numbers. We prove the complete $q$-order moment convergence for the partial sums of moving average processes $\Big \{X_n=\sum _{i=-\infty }^\infty a_i Y_{i+n},n\geq 1\Big \}$ based on the sequence $\{Y_i, -\infty <i<\infty \}$ of $\varphi $-mixing random variables under some suitable conditions. These results generalize and complement earlier results.
References:
[1] Baek, J.-I., Kim, T.-S., Liang, H.-Y.: On the convergence of moving average processes under dependent conditions. Aust. N. Z. J. Stat. 45 (2003), 331-342. DOI 10.1111/1467-842X.00287 | MR 1999515 | Zbl 1082.60028
[2] Burton, R. M., Dehling, H.: Large deviations for some weakly dependent random processes. Stat. Probab. Lett. 9 (1990), 397-401. DOI 10.1016/0167-7152(90)90031-2 | MR 1060081 | Zbl 0699.60016
[3] Chen, P., Hu, T.-C., Volodin, A.: Limiting behaviour of moving average processes under negative association assumption. Theory Probab. Math. Stat. 77 (2008), 165-176; Teor. Jmovirn. Mat. Stat. 77 149-160 (2007). DOI 10.1090/S0094-9000-09-00755-8 | MR 2432780 | Zbl 1199.60074
[4] Chen, P., Hu, T.-C., Volodin, A.: Limiting behaviour of moving average processes under $\varphi$-mixing assumption. Stat. Probab. Lett. 79 (2009), 105-111. DOI 10.1016/j.spl.2008.07.026 | MR 2483402 | Zbl 1154.60026
[5] Chen, P., Wang, D.: Convergence rates for probabilities of moderate deviations for moving average processes. Acta Math. Sin., Engl. Ser. 24 (2008), 611-622. DOI 10.1007/s10114-007-6062-7 | MR 2393155 | Zbl 1159.60015
[6] Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math., Acad. Sin. 16 (1988), 177-201. MR 1089491 | Zbl 0655.60028
[7] Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. DOI 10.1073/pnas.33.2.25 | MR 0019852 | Zbl 0030.20101
[8] Ibragimov, I. A.: Some limit theorems for stationary processes. Theor. Probab. Appl. 7 (1963), 349-382; Teor. Veroyatn. Primen 7 361-392 (1962), Russian. DOI 10.1137/1107036 | MR 0148125
[9] Kim, T.-S., Ko, M.-H.: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 78 (2008), 839-846. DOI 10.1016/j.spl.2007.09.009 | MR 2398357 | Zbl 1140.60315
[10] Lai, T. L.: Convergence rates and $r$-quick versions of the strong law for stationary mixing sequences. Ann. Probab. 5 (1977), 693-706. DOI 10.1214/aop/1176995713 | MR 0471043 | Zbl 0389.60020
[11] Li, D., Rao, M. Bhaskara, Wang, X.: Complete convergence of moving average processes. Stat. Probab. Lett. 14 (1992), 111-114. DOI 10.1016/0167-7152(92)90073-E | MR 1173407
[12] Li, D., Spătaru, A.: Refinement of convergence rates for tail probabilities. J. Theor. Probab. 18 (2005), 933-947. DOI 10.1007/s10959-005-7534-2 | MR 2289939 | Zbl 1085.60013
[13] Li, Y.-X., Zhang, L.-X.: Complete moment convergence of moving-average processes under dependence assumptions. Stat. Probab. Lett. 70 (2004), 191-197. DOI 10.1016/j.spl.2004.10.003 | MR 2108085 | Zbl 1056.62100
[14] Peligrad, M.: The $r$-quick version of the strong law for stationary $\varphi$-mixing sequences. Almost Everywhere Convergence, Proc. Int. Conf., Columbus/OH 1988 G. A. Edgar et al. Academic Press New York (1989), 335-348. MR 1035254
[15] Shao, Q.: A moment inequality and its applications. Acta Math. Sin. 31 (1988), 736-747 Chinese. MR 1000416 | Zbl 0698.60025
[16] Shao, Q.: Complete convergence for $\rho$-mixing sequences. Acta Math. Sin. 32 (1989), 377-393 Chinese. MR 1212051 | Zbl 0686.60025
[17] Yu, D., Wang, Z.: Complete convergence of moving average processes under negative dependence assumptions. Math. Appl. 15 (2002), 30-34. MR 1889135 | Zbl 1010.62081
[18] Zhang, L.-X.: Complete convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 30 (1996), 165-170. DOI 10.1016/0167-7152(95)00215-4 | MR 1417003 | Zbl 0873.60019
[19] Zhou, X.: Complete moment convergence of moving average processes under $\varphi$-mixing assumptions. Stat. Probab. Lett. 80 (2010), 285-292. DOI 10.1016/j.spl.2009.10.018 | MR 2593564 | Zbl 1186.60031
Partner of
EuDML logo