IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i12p2114-d841517.html
   My bibliography  Save this article

On De la Peña Type Inequalities for Point Processes

Author

Listed:
  • Naiqi Liu

    (School of Mathematics, Shandong University, Jinan 250100, China)

  • Vladimir V. Ulyanov

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Institute for Financial Studies, Shandong University, Jinan 250100, China)

  • Hanchao Wang

    (Institute for Financial Studies, Shandong University, Jinan 250100, China)

Abstract

There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Peña’s inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doléans–Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in Λ − coalescent.

Suggested Citation

  • Naiqi Liu & Vladimir V. Ulyanov & Hanchao Wang, 2022. "On De la Peña Type Inequalities for Point Processes," Mathematics, MDPI, vol. 10(12), pages 1-13, June.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2114-:d:841517
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/12/2114/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/10/12/2114/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Khoshnevisan, Davar, 1996. "Deviation inequalities for continuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 65(1), pages 17-30, December.
    2. Wang, Hanchao & Lin, Zhengyan & Su, Zhonggen, 2019. "On Bernstein type inequalities for stochastic integrals of multivariate point processes," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1605-1621.
    3. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    4. Dzhaparidze, K. & van Zanten, J. H., 2001. "On Bernstein-type inequalities for martingales," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 109-117, May.
    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. Alexander N. Tikhomirov & Vladimir V. Ulyanov, 2023. "On the Special Issue “Limit Theorems of Probability Theory”," Mathematics, MDPI, vol. 11(17), pages 1-4, August.

    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. Fan, Xiequan & Grama, Ion & Liu, Quansheng, 2012. "Hoeffding’s inequality for supermartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3545-3559.
    2. Pepin, Bob, 2021. "Concentration inequalities for additive functionals: A martingale approach," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 103-138.
    3. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    4. Fan, Xiequan & Alquier, Pierre & Doukhan, Paul, 2022. "Deviation inequalities for stochastic approximation by averaging," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 452-485.
    5. Fan, Xiequan, 2017. "Self-normalized deviation inequalities with application to t-statistic," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 158-164.
    6. Chen Li & Yuping Song, 2023. "Exponential Inequality of Marked Point Processes," Mathematics, MDPI, vol. 11(4), pages 1-11, February.
    7. Michael Diether, 2012. "Wavelet estimation in diffusions with periodicity," Statistical Inference for Stochastic Processes, Springer, vol. 15(3), pages 257-284, October.
    8. Reinhard Höpfner & Yury Kutoyants, 2010. "Estimating discontinuous periodic signals in a time inhomogeneous diffusion," Statistical Inference for Stochastic Processes, Springer, vol. 13(3), pages 193-230, October.
    9. Rio, Emmanuel, 2017. "New deviation inequalities for martingales with bounded increments," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1637-1648.
    10. Bu, Ruijun & Kim, Jihyun & Wang, Bin, 2023. "Uniform and Lp convergences for nonparametric continuous time regressions with semiparametric applications," Journal of Econometrics, Elsevier, vol. 235(2), pages 1934-1954.
    11. Sason, Igal, 2013. "Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1928-1936.
    12. Dedecker, Jérôme & Fan, Xiequan, 2015. "Deviation inequalities for separately Lipschitz functionals of iterated random functions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 60-90.
    13. Christensen, Kim & Oomen, Roel & Renò, Roberto, 2022. "The drift burst hypothesis," Journal of Econometrics, Elsevier, vol. 227(2), pages 461-497.
    14. Löcherbach, Eva & Orlandi, Enza, 2011. "Neighborhood radius estimation for variable-neighborhood random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2151-2185, September.
    15. Ruijun Bu & Jihyun Kim & Bin Wang, 2020. "Uniform and Lp Convergences of Nonparametric Estimation for Diffusion Models," Working Papers 202021, University of Liverpool, Department of Economics.

    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:gam:jmathe:v:10:y:2022:i:12:p:2114-:d:841517. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.