IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v16y2014i1d10.1007_s11009-012-9305-8.html
   My bibliography  Save this article

On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

Author

Listed:
  • Zhongquan Tan

    (Jiaxing University)

  • Enkelejd Hashorva

    (University of Lausanne)

Abstract

With motivation from Hüsler (Extremes 7:179–190, 2004) and Piterbarg (Extremes 7:161–177, 2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum.

Suggested Citation

  • Zhongquan Tan & Enkelejd Hashorva, 2014. "On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 169-185, March.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:1:d:10.1007_s11009-012-9305-8
    DOI: 10.1007/s11009-012-9305-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-012-9305-8
    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/s11009-012-9305-8?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. Hüsler, Jürg & Piterbarg, Vladimir, 2004. "Limit theorem for maximum of the storage process with fractional Brownian motion as input," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 231-250, December.
    2. Debicki, Krzysztof, 2002. "Ruin probability for Gaussian integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 151-174, March.
    3. James, Barry & James, Kang & Qi, Yongcheng, 2007. "Limit distribution of the sum and maximum from multivariate Gaussian sequences," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 517-532, March.
    4. Enkelejd Hashorva & Jürg Hüsler, 2000. "Extremes of Gaussian Processes with Maximal Variance near the Boundary Points," Methodology and Computing in Applied Probability, Springer, vol. 2(3), pages 255-269, September.
    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. Hüsler, Jürg & Zhang, Yueming, 2008. "On first and last ruin times of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1230-1235, August.
    2. Mladenovic, Pavle, 2009. "Maximum of a partial sample in the uniform AR(1) processes," Statistics & Probability Letters, Elsevier, vol. 79(11), pages 1414-1420, June.
    3. Gong, Siliang & Zhang, Kai & Liu, Yufeng, 2018. "Efficient test-based variable selection for high-dimensional linear models," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 17-31.
    4. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2014. "On the probability of conjunctions of stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 141-148.
    5. Dëbicki, Krzysztof & Kisowski, Pawel, 2008. "A note on upper estimates for Pickands constants," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2046-2051, October.
    6. Mladenovic, Pavle & Piterbarg, Vladimir, 2006. "On asymptotic distribution of maxima of complete and incomplete samples from stationary sequences," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1977-1991, December.
    7. Chengxiu Ling & Hong Zhang, 2020. "On Generalized Berman Constants," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1125-1143, September.
    8. Long Bai & Krzysztof Dȩbicki & Enkelejd Hashorva & Li Luo, 2018. "On Generalised Piterbarg Constants," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 137-164, March.
    9. Peng, Zuoxiang & Cao, Lunfeng & Nadarajah, Saralees, 2010. "Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2641-2647, November.
    10. Krzysztof Dȩbicki & Zbigniew Michna & Xiaofan Peng, 2019. "Approximation of Sojourn Times of Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1183-1213, December.
    11. Dieker, A.B., 2005. "Extremes of Gaussian processes over an infinite horizon," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 207-248, February.
    12. Jinhui Guo & Yingyin Lu, 2023. "Joint behavior of point processes of clusters and partial sums for stationary bivariate Gaussian triangular arrays," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 17-37, February.
    13. Tan, Zhongquan & Hashorva, Enkelejd, 2013. "Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2983-2998.
    14. Aiping Hu & Zuoxiang Peng & Yongcheng Qi, 2009. "Joint behavior of point process of exceedances and partial sum from a Gaussian sequence," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 279-295, November.

    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:metcap:v:16:y:2014:i:1:d:10.1007_s11009-012-9305-8. 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.