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 [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.