On probabilistic convergence rates of stochastic Bernstein polynomials
On probabilistic convergence rates of stochastic Bernstein polynomials
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DOI：
10.1090/mcom/3589
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发表时间：
2020-11
期刊：
影响因子：
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通讯作者：
Xingping Sun;Zongmin Wu;Xuan Zhou
中科院分区：
文献类型：
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作者：
Xingping Sun;Zongmin Wu;Xuan Zhou
在这篇文章中，我们引入了随机Bernstein多项式的“-概率收敛”的概念，它建立在上的相同、独立和均匀分布的随机变量的顺序统计量上。我们根据目标函数的连续模建立了幂收敛速度和指数收敛速度。在此范围内，我们得到了相应概率收敛的高斯尾界。我们对案例的结果证实了第二和第三作者提出的猜想。蒙特卡罗模拟结果表明，本文所研究的随机Bernstein近似格式达到了与经典Bernstein近似格式相当的计算目标，并有力地表明了所证明的高斯尾界在这种情况下也是成立的。参考文献
In this article, we introduce the notion “-probabilistic convergence"() of stochastic Bernstein polynomials built upon order statistics of identically, independently, and uniformly distributed random variables on. We establish power and exponential convergence rates in terms of the modulus of continuity of a target function. Forin the rangewe obtain Gaussian tail bounds for the corresponding probabilistic convergence. Our result for the caseconfirms a conjecture raised by the second and third authors. Monte Carlo simulations (presented at the end of the article) show that the stochastic Bernstein approximation scheme studied herein achieves comparable computational goals to the classical Bernstein approximation, and indicate strongly that the Gaussian tail bounds proved foralso hold true for the cases. References