Chebyshev type inequality for stochastic Bernstein polynomials
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- by Xingping Sun and Zongmin Wu PDF
- Proc. Amer. Math. Soc. 147 (2019), 671-679 Request permission
Abstract:
We study a class of stochastic Bernstein polynomials built from order statistics of identically, independently, and uniformly distributed random variables on $[0,1]$. We establish a Chebyshev type inequality for the probabilistic convergence of a stochastic Bernstein polynomial sequence to its target function. This is a major improvement of the main result of Wu, Sun, and Ma [Adv. Comput. Math. 38 (2013), no. 1, pp. 187–205]. Moreover, the method we develop here in dealing with varying-weighted sums of dependent random variables is of independent interest.References
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Additional Information
- Xingping Sun
- Affiliation: College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, China, and Department of Mathematics, Missouri State University, Springfield, Missouri 65897
- MR Author ID: 270544
- Zongmin Wu
- Affiliation: Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Science, Fudan University, Shanghai, China
- MR Author ID: 268328
- Received by editor(s): September 12, 2017
- Received by editor(s) in revised form: March 11, 2018
- Published electronically: October 31, 2018
- Additional Notes: The work described in this paper was partially supported by the National Natural Science Foundation of China under Grant No. 11461161006 and by a grant from the Research Grants Council of Hong Kong [Project No. CityU 104012].
- Communicated by: Yuan Xu
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 671-679
- MSC (2010): Primary 41A25, 41A63, 42B08
- DOI: https://doi.org/10.1090/proc/14161
- MathSciNet review: 3894906