IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v29y2016i2d10.1007_s10959-014-0591-7.html
   My bibliography  Save this article

The Cameron–Martin Theorem for (p-)Slepian Processes

Author

Listed:
  • Wolfgang Bischoff

    (Catholic University of Eichstätt-Ingolstadt)

  • Andreas Gegg

    (Catholic University of Eichstätt-Ingolstadt)

Abstract

We show a Cameron–Martin theorem for Slepian processes $$W_t:=\frac{1}{\sqrt{p}}(B_t-B_{t-p}), t\in [p,1]$$ W t : = 1 p ( B t - B t - p ) , t ∈ [ p , 1 ] , where $$p\ge \frac{1}{2}$$ p ≥ 1 2 and $$B_s$$ B s is Brownian motion. More exactly, we determine the class of functions $$F$$ F for which a density of $$F(t)+W_t$$ F ( t ) + W t with respect to $$W_t$$ W t exists. Moreover, we prove an explicit formula for this density. p-Slepian processes are closely related to Slepian processes. p-Slepian processes play a prominent role among others in scan statistics and in testing for parameter constancy when data are taken from a moving window.

Suggested Citation

  • Wolfgang Bischoff & Andreas Gegg, 2016. "The Cameron–Martin Theorem for (p-)Slepian Processes," Journal of Theoretical Probability, Springer, vol. 29(2), pages 707-715, June.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:2:d:10.1007_s10959-014-0591-7
    DOI: 10.1007/s10959-014-0591-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-014-0591-7
    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/s10959-014-0591-7?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. Wolfgang Bischoff & Frank Miller, 2000. "Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 658-679, December.
    2. Liu, Jin V. & Huang, Zongfu & Mao, Hongjun, 2014. "Karhunen–Loève expansion for additive Slepian processes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 93-99.
    3. Bischoff, Wolfgang & Hashorva, Enkelejd, 2005. "A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 265-271, October.
    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. 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.
    2. Deng, Pingjin, 2017. "Boundary non-crossing probabilities for Slepian process," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 28-35.
    3. Enkelejd Hashorva, 2010. "Boundary Non-crossings of Brownian Pillow," Journal of Theoretical Probability, Springer, vol. 23(1), pages 193-208, March.
    4. Enkelejd Hashorva & Yuliya Mishura & Georgiy Shevchenko, 2021. "Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics," Journal of Theoretical Probability, Springer, vol. 34(2), pages 728-754, June.
    5. Friede, T. & Miller, F. & Bischoff, W. & Kieser, M., 2001. "A note on change point estimation in dose-response trials," Computational Statistics & Data Analysis, Elsevier, vol. 37(2), pages 219-232, August.
    6. Bischoff, Wolfgang & Gegg, Andreas, 2016. "Boundary crossing probabilities for (q,d)-Slepian-processes," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 139-144.
    7. Bischoff, Wolfgang & Hashorva, Enkelejd & Hüsler, Jürg & Miller, Frank, 2004. "On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 105-115, January.
    8. Pingjin Deng, 2016. "Asymptotic of Non-Crossings probability of Additive Wiener Fields," Papers 1610.07131, arXiv.org.

    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:29:y:2016:i:2:d:10.1007_s10959-014-0591-7. 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.