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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
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    References listed on IDEAS

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    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.
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    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.

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