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Weak invariance principles for sums of dependent random functions

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  • Berkes, István
  • Horváth, Lajos
  • Rice, Gregory

Abstract

Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure.

Suggested Citation

  • Berkes, István & Horváth, Lajos & Rice, Gregory, 2013. "Weak invariance principles for sums of dependent random functions," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 385-403.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:385-403
    DOI: 10.1016/j.spa.2012.10.003
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    References listed on IDEAS

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    1. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    2. Aue, Alexander & Hörmann, Siegfried & Horváth, Lajos & Hušková, Marie & Steinebach, Josef G., 2012. "Sequential Testing For The Stability Of High-Frequency Portfolio Betas," Econometric Theory, Cambridge University Press, vol. 28(4), pages 804-837, August.
    3. Dedecker, Jérôme & Merlevède, Florence, 2003. "The conditional central limit theorem in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 229-262, December.
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    Cited by:

    1. Lajos Horváth & Gregory Rice, 2015. "Testing Equality Of Means When The Observations Are From Functional Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(1), pages 84-108, January.
    2. Horváth, Lajos & Hušková, Marie & Rice, Gregory, 2013. "Test of independence for functional data," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 100-119.
    3. Salish, Nazarii & Gleim, Alexander, 2019. "A moment-based notion of time dependence for functional time series," Journal of Econometrics, Elsevier, vol. 212(2), pages 377-392.
    4. Górecki, Tomasz & Horváth, Lajos & Kokoszka, Piotr, 2018. "Change point detection in heteroscedastic time series," Econometrics and Statistics, Elsevier, vol. 7(C), pages 63-88.
    5. Horváth, Lajos & Rice, Gregory & Zhao, Yuqian, 2022. "Change point analysis of covariance functions: A weighted cumulative sum approach," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    6. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    7. Liu, Xialu & Xiao, Han & Chen, Rong, 2016. "Convolutional autoregressive models for functional time series," Journal of Econometrics, Elsevier, vol. 194(2), pages 263-282.
    8. Horváth, Lajos & Rice, Gregory, 2015. "Testing for independence between functional time series," Journal of Econometrics, Elsevier, vol. 189(2), pages 371-382.
    9. Chen, Yichao & Pun, Chi Seng, 2019. "A bootstrap-based KPSS test for functional time series," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    10. Dehling, Herold & Sharipov, Olimjon Sh. & Wendler, Martin, 2015. "Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 200-215.
    11. Degui Li & Peter M. Robinson & Han Lin Shang, 2021. "Local Whittle estimation of long‐range dependence for functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 685-695, September.
    12. Burdejova, P. & Härdle, W. & Kokoszka, P. & Xiong, Q., 2017. "Change point and trend analyses of annual expectile curves of tropical storms," Econometrics and Statistics, Elsevier, vol. 1(C), pages 101-117.
    13. Horváth, Lajos & Kokoszka, Piotr & Rice, Gregory, 2014. "Testing stationarity of functional time series," Journal of Econometrics, Elsevier, vol. 179(1), pages 66-82.
    14. Cerovecki, Clément & Hörmann, Siegfried, 2017. "On the CLT for discrete Fourier transforms of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 282-295.
    15. Holger Dette & Kevin Kokot & Stanislav Volgushev, 2020. "Testing relevant hypotheses in functional time series via self‐normalization," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 629-660, July.

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