IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v108y2003i2p229-262.html
   My bibliography  Save this article

The conditional central limit theorem in Hilbert spaces

Author

Listed:
  • Dedecker, Jérôme
  • Merlevède, Florence

Abstract

In this paper, we give necessary and sufficient conditions for a stationary sequence of random variables with values in a separable Hilbert space to satisfy the conditional central limit theorem introduced in Dedecker and Merlevède (Ann. Probab. 30 (2002) 1044-1081). As a consequence, this theorem implies stable convergence of the normalized partial sums to a mixture of normal distributions. We also establish the functional version of this theorem. Next, we show that these conditions are satisfied for a large class of weakly dependent sequences, including strongly mixing sequences as well as mixingales. Finally, we present an application to linear processes generated by some stationary sequences of -valued random variables.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:108:y:2003:i:2:p:229-262
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00115-7
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Chen, Xiaohong & White, Halbert, 1998. "Central Limit And Functional Central Limit Theorems For Hilbert-Valued Dependent Heterogeneous Arrays With Applications," Econometric Theory, Cambridge University Press, vol. 14(2), pages 260-284, April.
    2. Kuelbs, J., 1973. "The invariance principle for Banach space valued random variables," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 161-172, June.
    3. Dedecker, Jérôme & Doukhan, Paul, 2003. "A new covariance inequality and applications," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 63-80, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Leucht, Anne, 2012. "Characteristic function-based hypothesis tests under weak dependence," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 67-89.
    2. 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.
    3. Jérôme Dedecker & Florence Merlevède & Dalibor Volný, 2007. "On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 20(4), pages 971-1004, December.
    4. Leão, Dorival & Ohashi, Alberto, 2012. "On the Discrete Cramér-von Mises Statistics under Random Censorship," Insper Working Papers wpe_275, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    5. Álvarez-Liébana, J. & Bosq, D. & Ruiz-Medina, M.D., 2017. "Asymptotic properties of a component-wise ARH(1) plug-in predictor," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 12-34.
    6. Álvarez-Liébana, Javier & Bosq, Denis & Ruiz-Medina, María D., 2016. "Consistency of the plug-in functional predictor of the Ornstein–Uhlenbeck process in Hilbert and Banach spaces," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 12-22.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Florence Merlevède, 2003. "On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation," Journal of Theoretical Probability, Springer, vol. 16(3), pages 625-653, July.
    2. Shao, Xiaofeng, 2011. "A bootstrap-assisted spectral test of white noise under unknown dependence," Journal of Econometrics, Elsevier, vol. 162(2), pages 213-224, June.
    3. R. Anton Braun & Huiyu Li & John Stachurski, 2012. "Generalized Look-Ahead Methods for Computing Stationary Densities," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 489-500, August.
    4. Escanciano, Juan Carlos & Jacho-Chávez, David T., 2010. "Approximating the critical values of Cramér-von Mises tests in general parametric conditional specifications," Computational Statistics & Data Analysis, Elsevier, vol. 54(3), pages 625-636, March.
    5. Chen Xiaohong & White Halbert, 2002. "Asymptotic Properties of Some Projection-based Robbins-Monro Procedures in a Hilbert Space," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 6(1), pages 1-55, April.
    6. Michael Messer, 2022. "Bivariate change point detection: Joint detection of changes in expectation and variance," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 886-916, June.
    7. Jirak, Moritz, 2012. "Change-point analysis in increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 136-159.
    8. Dong, Hao & Taylor, Luke, 2022. "Nonparametric Significance Testing In Measurement Error Models," Econometric Theory, Cambridge University Press, vol. 38(3), pages 454-496, June.
    9. Andrii Babii & Eric Ghysels & Jonas Striaukas, 2022. "Machine Learning Time Series Regressions With an Application to Nowcasting," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(3), pages 1094-1106, June.
    10. Marine Carrasco & Barbara Rossi, 2016. "In-Sample Inference and Forecasting in Misspecified Factor Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 313-338, July.
    11. Steffen Betsch & Bruno Ebner, 2019. "A new characterization of the Gamma distribution and associated goodness-of-fit tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(7), pages 779-806, October.
    12. Florens, Jean-Pierre & Simoni, Anna, 2012. "Nonparametric estimation of an instrumental regression: A quasi-Bayesian approach based on regularized posterior," Journal of Econometrics, Elsevier, vol. 170(2), pages 458-475.
    13. Babii, Andrii & Ball, Ryan T. & Ghysels, Eric & Striaukas, Jonas, 2023. "Machine learning panel data regressions with heavy-tailed dependent data: Theory and application," Journal of Econometrics, Elsevier, vol. 237(2).
    14. Won-Ki Seo, 2020. "Functional Principal Component Analysis for Cointegrated Functional Time Series," Papers 2011.12781, arXiv.org, revised Apr 2023.
    15. Paul Doukhan & Jean-David Fermanian & Gabriel Lang, 2009. "An empirical central limit theorem with applications to copulas under weak dependence," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 65-87, February.
    16. P. Chigansky & Yu. Kutoyants, 2013. "Estimation in threshold autoregressive models with correlated innovations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(5), pages 959-992, October.
    17. 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.
    18. Hélène Cossette & Etienne Marceau & Véronique Maume-Deschamps, 2011. "Adjustment Coefficient for Risk Processes in Some Dependent Contexts," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 695-721, December.
    19. Kokoszka, Piotr & Reimherr, Matthew, 2013. "Asymptotic normality of the principal components of functional time series," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1546-1562.
    20. Ngai Chan & Yury Kutoyants, 2012. "On parameter estimation of threshold autoregressive models," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 81-104, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:108:y:2003:i:2:p:229-262. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.