Skip to main content
SpringerLink
Log in

Complete Moment Convergence for the Dependent Linear Processes with Random Coefficients

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, A., Rosalsky, A.: Some general strong laws for weighted sums of stochastically dominated random variables. Stock. Anal. Appl., 5(1), 1–16 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin., 16(3), 177–201 (1988)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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

    Book  Google Scholar 

  7. 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)

    MathSciNet  Google Scholar 

  8. Kim, T. S., Ko, M. H.: Complete moment convergence of moving average processes under dependence assumptions. Statist. Probab. Lett., 78(7), 839–846 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kulik, R.: Limit theorems for moving averages with random coefficients and heavy-tailed noise. J. Appl. Probab., 43(1), 245–256 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist., 1137–1153 (1966)

    Google Scholar 

  15. Liu, L.: Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math., 53(6), 1421–1434 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Louhichi, S., Soulier, P.: Marcinkiewicz-Zygmund strong laws for infinite variance time series. Stat. Inference Stoch. Process., 3(1), 31–40 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. McLeish, D. L.: Invariance principles for dependent variables. Probab. Theory Related Fields, 32(3), 165178 (1975)

    MathSciNet  Google Scholar 

  18. McLeish, D. L.: Maximal inequality and dependent strong laws. Ann. Probab., 3(5), 829–839 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Phillips, P. C., Solo, S.: Asymptotics for linear processes. Ann. Statist., 971–1001 (1992)

    Google Scholar 

  20. Qiu, D., Chen, P.: Complete moment convergence for iid random variables. Statist. Probab. Lett., 91, 76–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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

  22. Saavedra, P., et al.: Estimation of population spectrum for linear processes with random coefficients. Comput. Statist., 23(1), 79–98 (2008)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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

  25. Sung, S. H.: Convergence of moving average processes for dependent random variables. Comm. Statist. Theory Methods, 40(13), 2366–2376 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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

  27. 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

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    MathSciNet  Google Scholar 

  33. 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

  34. 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)

  35. Yang, W., Hu, S.: Complete moment convergence of pairwise NQD random variables. Stochastics, 87(2), 199–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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

  37. Zhang, S., Qu, C., Sun, F.: On the Complete Convergence for END Random Variables. Appl. Math. Sci., 12(4), 161–173 (2018)

    Google Scholar 

  38. Zhou, X.: Complete moment convergence of moving average processes under φ-mixing assumptions. Statist. Probab. Lett., 80(5–6), 285–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, Iran

    Seyed Mohammad Hosseini & Ahmad Nezakati

Authors
  1. Seyed Mohammad Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmad Nezakati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmad Nezakati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-019-8205-z

Keywords

MR(2010) Subject Classification

Search

Navigation