Abstract
In this paper, we investigate the complete moment convergence for dependent linear processes with random coefficients to form \({X_t} = \sum\nolimits_{j = - \infty}^\infty {{A_j}{\varepsilon _{t - j}}} \), where {εn,n ∈ ℤ} is a sequence of END stochastically dominated random variables and {An,n ∈ ℤ} is a sequence of random varibles. As applications, the convergence rate, Marcinkiewicz-Zygmund strong law and strong law of large numbers for this linear process are established.
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Adler, A., Rosalsky, A.: Some general strong laws for weighted sums of stochastically dominated random variables. Stock. Anal. Appl., 5(1), 1–16 (1987)
Adler, A., Rosalsky, A., Taylor, R. L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Int. J. Math. Math. Sci., 12(3), 507–529 (1989)
Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin., 16(3), 177–201 (1988)
Ding, Y., Tang, X., Deng, X., et al.: Complete moment convergence for weighted sums of extended negatively dependent random variables. Filomat, 31(14), 4341–4352 (2017)
Guo, M., Zhu, D., Ren, Y.: Complete q th moment convergence of weighted sums for arrays of rowwise negatively associated random variables. Stochastics, 87(2), 257–272 (2015)
Hosseini, S. H., Nezakati, A.: Convergence rates in the law of large numbers for END linear processes with random coefficients, Comm. Statist. Theory Methods, 2018
Kim, H. C.: On the complete moment convergence of moving average processes generated by iid random variables. Korean Ann. Math., 25(1), 7–17 (2008)
Kim, T. S., Ko, M. H.: Complete moment convergence of moving average processes under dependence assumptions. Statist. Probab. Lett., 78(7), 839–846 (2008)
Kim, T. S., Ko, M. H., Choi, Y. K.: Complete moment convergence of moving average processes with dependent innovations. J. Korean Math. Soc., 45(2), 355–365 (2008)
Ko, M. H.: Strong laws of large numbers for linear processes generated by associated random variables in a Hilbert space. Honam Math. J., 30(4), 703–711 (2008)
Ko, M. H., Kim, T. S., Ryu, D. H.: On the complete moment convergence of moving average processes generated by ρ*-mixing sequences. Commun. Korean Math. Soc., 23(4), 597–606 (2008)
Kulik, R.: Limit theorems for moving averages with random coefficients and heavy-tailed noise. J. Appl. Probab., 43(1), 245–256 (2006)
Le Guo, M.: Equivalent conditions of complete moment convergence of weighted sums for φ-mixing sequence of random variables. Comm. Statist. Theory Methods, 43(10–12), 2527–2539 (2014)
Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist., 1137–1153 (1966)
Liu, L.: Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math., 53(6), 1421–1434 (2010)
Louhichi, S., Soulier, P.: Marcinkiewicz-Zygmund strong laws for infinite variance time series. Stat. Inference Stoch. Process., 3(1), 31–40 (2000)
McLeish, D. L.: Invariance principles for dependent variables. Probab. Theory Related Fields, 32(3), 165178 (1975)
McLeish, D. L.: Maximal inequality and dependent strong laws. Ann. Probab., 3(5), 829–839 (1975)
Phillips, P. C., Solo, S.: Asymptotics for linear processes. Ann. Statist., 971–1001 (1992)
Qiu, D., Chen, P.: Complete moment convergence for iid random variables. Statist. Probab. Lett., 91, 76–82 (2014)
Qiu, D., Chen, P.: Complete moment convergence for product sums of sequence of extended negatively dependent random variables. J. Inequal. Appl., 2015:212, 15 pp., doi: https://doi.org/10.1186/s13660-015-0730-4
Saavedra, P., et al.: Estimation of population spectrum for linear processes with random coefficients. Comput. Statist., 23(1), 79–98 (2008)
Shen, A.: Complete convergence for weighted sums of END random variables and its application to non-parametric regression models. J. Nonparametr. Stat., 28(4), 702–715 (2016)
Sung, S. H.: Moment inequalities and complete moment convergence. J. Inequal. Appl., Article ID 271265, 14 pp., doi: https://doi.org/10.1155/2009/271265
Sung, S. H.: Convergence of moving average processes for dependent random variables. Comm. Statist. Theory Methods, 40(13), 2366–2376 (2011)
Sung, S. H.: Complete q th moment convergence for arrays of random variables. J. Inequal. Appl., 2013:24, 11 pp, doi:https://doi.org/10.1186/1029-242X-2013-24
Wang, J., Wu, Q.: Central limit theorem for stationary linear processes generated by linearly negative quadrant-dependent sequence. J. Inequal. Appl., 2012:45, 7 pp., doi:https://doi.org/10.1186/1029-242X-2012-45
Wang, S., Wang, X.: Precise large deviations for random sums of END real-valued random variables with consistent variation. J. Math. Anal. Appl., 402(2), 660–667 (2013)
Wang, X., Shen, A., Li, X.: A note on complete convergence of weighted sums for array of rowwise AANA random variables. J. Inequal. Appl., 2013:359., 13 pp., doi:https://doi.org/10.1186/1029-242X-2013-359
Wang, X., Li, X., Hu, S.: Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables. Appl. Math., 59(5), 589–607 (2014)
Wang, X., et al.: Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics, 49(2), 396–407 (2015)
Wu, Y. F., Cabrea, M. O., Volodin, A.: Complete convergence and complete moment convergence for arrays of rowwise END random variables. Glas. Mat., 49(69), 449–468 (2014)
Yang, W., et al.: On complete convergence of moving average process for AANA sequence. Discrete Dyn. Nat. Soc., Article ID315138, 15 pp., doi:https://doi.org/10.1155/2012/863931
Yang, W., et al.: The convergence of double-indexed weighted sums of martingale differences and its application. Abstr. Appl. Anal., 2014, Article ID 893906 (2014)
Yang, W., Hu, S.: Complete moment convergence of pairwise NQD random variables. Stochastics, 87(2), 199–208 (2015)
Zhang, G.: Complete convergence for Sungs type weighted sums of END random variables. J. Inequal. Appl., Article ID 383805, 8 pp., doi:https://doi.org/10.1155/2010/383805
Zhang, S., Qu, C., Sun, F.: On the Complete Convergence for END Random Variables. Appl. Math. Sci., 12(4), 161–173 (2018)
Zhou, X.: Complete moment convergence of moving average processes under φ-mixing assumptions. Statist. Probab. Lett., 80(5–6), 285–292 (2010)
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We would like to thank the referee for the constructive and substantial comments which greatly improved the presentation and led to put many details in the paper.
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Hosseini, S.M., Nezakati, A. Complete Moment Convergence for the Dependent Linear Processes with Random Coefficients. Acta. Math. Sin.-English Ser. 35, 1321–1333 (2019). https://doi.org/10.1007/s10114-019-8205-z
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DOI: https://doi.org/10.1007/s10114-019-8205-z
Keywords
- Complete moment convergence
- extended negatively dependent random variables
- linear processes
- random coefficients