Chebyshev type inequality for stochastic Bernstein polynomials
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. Moreover, the method we develop here in dealing with varying-weighted sums of dependent random variables is of independent interest.
Sun, Xingping, and Zongmin Wu. "Chebyshev type inequality for stochastic Bernstein polynomials." Proceedings of the American Mathematical Society 147, no. 2 (2019): 671-679.
Proceedings of the American Mathematical Society