Chebyshev type inequality for stochastic Bernstein polynomials

Abstract

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.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1090/proc/14161

Keywords

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

Publication Date

2018

Journal Title

Proceedings of the American Mathematical Society

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