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.
Bernstein inequality, Bernstein polynomial, Chebyshev inequality, Modulus of continuity, Stochastic Bernstein polynomial
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