Abstract
Recently, authors of Wu et al. (Adv. Comput. Math. 38:187-205, 2013) studied a Bernstein polynomial approximation scheme based on stochastic sampling and obtained a sixth-order moment estimate for the underlying random variable in terms of the modulus of continuity. In the current paper, we employ a new technique and establish estimates for all the even-order moments. Our work gives a strong indication that the probabilistic convergence rate of the stochastic Bernstein approximation is exponential with respect to the modulus of continuity, which we leave as a conjecture at the end of the paper.
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Funding
This work is financially supported by the NSFC (11631015,91330201), joint Research Fund by National Science Foundation of China and Research Grant Council of Hong Kong (11461161006).
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Communicated by: Tomas Sauer
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Wu, Z., Zhou, X. Polynomial convergence order of stochastic Bernstein approximation. Adv Comput Math 46, 8 (2020). https://doi.org/10.1007/s10444-020-09742-w
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DOI: https://doi.org/10.1007/s10444-020-09742-w