IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v36y2023i1d10.1007_s10959-022-01177-x.html
   My bibliography  Save this article

Asymptotic Normality in Banach Spaces via Lindeberg Method

Author

Listed:
  • Alfredas Račkauskas

    (Vilnius University)

  • Charles Suquet

    (Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé)

Abstract

The relation between weak convergence of probabilities on a smooth Banach space and uniform convergence over a certain class of smooth functions is established. This leads to an extension of Lindeberg’s proof of the central limit theorem in a Banach space framework. As a result, asymptotic normality is proved for sums of Banach space random variables including triangular arrays and weighted linear processes.

Suggested Citation

  • Alfredas Račkauskas & Charles Suquet, 2023. "Asymptotic Normality in Banach Spaces via Lindeberg Method," Journal of Theoretical Probability, Springer, vol. 36(1), pages 409-455, March.
  • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-022-01177-x
    DOI: 10.1007/s10959-022-01177-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-022-01177-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-022-01177-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Cremers, Heinz & Kadelka, Dieter, 1986. "On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 305-317, February.
    2. Marwa Banna & Florence Merlevède, 2015. "Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 745-783, June.
    3. Banna, Marwa & Merlevède, Florence & Peligrad, Magda, 2015. "On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2700-2726.
    Full references (including those not matched with items on IDEAS)

    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. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    2. Karim M. Abadir & Walter Distaso & Liudas Giraitis, 2011. "An I() model with trend and cycles," Post-Print hal-00834425, HAL.
    3. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2015. "Joint aggregation of random-coefficient AR(1) processes with common innovations," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 73-82.
    4. Jamshid Namdari & Debashis Paul & Lili Wang, 2021. "High-Dimensional Linear Models: A Random Matrix Perspective," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 645-695, August.
    5. Aue, Alexander & Van Delft, Anne, 2017. "Testing for stationarity of functional time series in the frequency domain," LIDAM Discussion Papers ISBA 2017001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Höpfner Reinhard & Kutoyants Yury A., 2009. "On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion," Statistics & Risk Modeling, De Gruyter, vol. 27(4), pages 309-326, December.
    7. James Cameron & Pramita Bagchi, 2022. "A test for heteroscedasticity in functional linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 519-542, June.
    8. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2011. "An I(d) model with trend and cycles," Journal of Econometrics, Elsevier, vol. 163(2), pages 186-199, August.
    9. Heiny, Johannes & Mikosch, Thomas, 2018. "Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2779-2815.
    10. Mathias Mørck Ljungdahl & Mark Podolskij, 2020. "A minimal contrast estimator for the linear fractional stable motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 381-413, July.
    11. Oliveira, P. E. & Suquet, Ch., 1998. "Weak convergence in Lp(0,1) of the uniform empirical process under dependence," Statistics & Probability Letters, Elsevier, vol. 39(4), pages 363-370, August.
    12. Merlevède, F. & Peligrad, M., 2016. "On the empirical spectral distribution for matrices with long memory and independent rows," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2734-2760.
    13. Heiny, Johannes & Mikosch, Thomas, 2021. "Large sample autocovariance matrices of linear processes with heavy tails," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 344-375.
    14. Düker, Marie-Christine, 2018. "Limit theorems for Hilbert space-valued linear processes under long range dependence," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1439-1465.
    15. Sanders, Jaron & Van Werde, Alexander, 2023. "Singular value distribution of dense random matrices with block Markovian dependence," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 453-504.
    16. Characiejus, Vaidotas & Račkauskas, Alfredas, 2014. "Operator self-similar processes and functional central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2605-2627.
    17. Höpfner, R. & Löcherbach, E., 1999. "On local asymptotic normality for birth and death on a flow," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 61-77, September.
    18. Fleermann, Michael & Kirsch, Werner & Kriecherbauer, Thomas, 2021. "The almost sure semicircle law for random band matrices with dependent entries," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 172-200.
    19. Jun, Sung Jae & Pinkse, Joris & Wan, Yuanyuan, 2015. "Classical Laplace estimation for n3-consistent estimators: Improved convergence rates and rate-adaptive inference," Journal of Econometrics, Elsevier, vol. 187(1), pages 201-216.
    20. A. Lytova, 2018. "Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1024-1057, June.

    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:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-022-01177-x. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.