Abstract: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 . For in the range we obtain Gaussian tail bounds for the corresponding probabilistic convergence. Our result for the case confirms 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 for also hold true for the cases .

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中文翻译：

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随机伯恩斯坦多项式的概率收敛速度

摘要：本文介绍了随机伯恩斯坦多项式的``-概率收敛'' （）概念，该概念建立在随机，均匀和均匀分布的随机变量的阶统计基础之上，并根据模数建立幂和指数收敛速率目标函数的连续性。在此范围内，我们获得了对应概率收敛的高斯尾边界。 确认第二和第三作者提出的猜想。蒙特卡洛模拟（在本文的结尾给出）表明，本文研究的随机伯恩斯坦逼近方案达到了与经典伯恩斯坦逼近相当的计算目标，并强烈表明证明高斯尾定界在这种情况下也成立。

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