IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i6p2353-2382.html
   My bibliography  Save this article

Matrix normalized convergence of a Lévy process to normality at zero

Author

Listed:
  • Maller, Ross A.
  • Mason, David M.

Abstract

We give a necessary and sufficient condition for a d-dimensional Lévy process to be in the matrix normalized domain of attraction of a d-dimensional normal random vector, as t↓0. This transfers to the Lévy case classical results of Feller, Khinchin, Lévy and Hahn and Klass for random walks. A specific construction of the norming matrix is given, and it is shown that centering constants may be taken as 0. Functional and self-normalization results are also given, as is a necessary and sufficient condition for the process to be in the matrix normalized domain of partial attraction of the normal.

Suggested Citation

  • Maller, Ross A. & Mason, David M., 2015. "Matrix normalized convergence of a Lévy process to normality at zero," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2353-2382.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:6:p:2353-2382
    DOI: 10.1016/j.spa.2015.01.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915000149
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.01.003?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. Maller, R. A., 1993. "Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 191-219, February.
    2. Vu, H. T. V. & Maller, R. A. & Klass, M. J., 1996. "On the Studentisation of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 142-155, April.
    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. Martsynyuk, Yuliya V., 2013. "On the generalized domain of attraction of the multivariate normal law and asymptotic normality of the multivariate Student t-statistic," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 402-411.
    2. Kesten, Harry & Maller, R. A., 1997. "Random Deletion Does Not Affect Asymptotic Normality or Quadratic Negligibility," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 136-179, October.
    3. Csörgő, Miklós & Martsynyuk, Yuliya V., 2011. "Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2925-2953.
    4. Martsynyuk, Yuliya V., 2012. "Invariance principles for a multivariate Student process in the generalized domain of attraction of the multivariate normal law," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2270-2277.
    5. Hongchang Hu & Weifu Hu & Xinxin Yu, 2021. "Pseudo-maximum likelihood estimators in linear regression models with fractional time series," Statistical Papers, Springer, vol. 62(2), pages 639-659, April.
    6. Mark M. Meerschaert & Hans-Peter Scheffler, 1999. "Sample Covariance Matrix for Random Vectors with Heavy Tails," Journal of Theoretical Probability, Springer, vol. 12(3), pages 821-838, July.
    7. H. Vu & R. Maller & X. Zhou, 1998. "Asymptotic Properties of a Class of Mixture Models for Failure Data: The Interior and Boundary Cases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 627-653, December.
    8. Maller, R. A., 2003. "Asymptotics of regressions with stationary and nonstationary residuals," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 33-67, May.
    9. Sepanski, Steven J., 1996. "Asymptotics for multivariate t-statistic for random vectors in the generalized domain of attraction of the multivariate normal law," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 179-188, October.
    10. Choi, K. C. & Zhou, X., 2002. "Large Sample Properties of Mixture Models with Covariates for Competing Risks," Journal of Multivariate Analysis, Elsevier, vol. 82(2), pages 331-366, August.

    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:125:y:2015:i:6:p:2353-2382. 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.