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On Bernstein type inequalities for stochastic integrals of multivariate point processes

Author

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  • Wang, Hanchao
  • Lin, Zhengyan
  • Su, Zhonggen

Abstract

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:5:p:1605-1621
    DOI: 10.1016/j.spa.2018.05.014
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    Cited by:

    1. Chen Li & Yuping Song, 2023. "Exponential Inequality of Marked Point Processes," Mathematics, MDPI, vol. 11(4), pages 1-11, February.
    2. 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.

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