Stochastic Quasi-Interpolation with Bernstein Polynomials

Abstract

We introduce the notion “stochastic quasi-interpolation on compact Hausdorff spaces”, and establish Gaussian-type Lp-concentration inequalities (1 ? p? ?) for stochastic Bernstein polynomials in terms of the modulus of continuity of a target function f? C[0 , 1]. For p in the range 1 ? p< ?, these inequalities hold true unconditionally in the sense that no additional assumption on a given target function is required. For the case p= ?, our proof calls for a crucial application of Dvoretzky–Kiefer–Wolfowitz inequality (Dvoretzky et al. in Ann Math Stat 27(3):642–669, 1956; Massart in Ann Probab 18(3):1269–1283, 1990) , and requires a moderate decay condition on the modulus of continuity. Our result for the case p= ? confirms a similar conjecture raised in Sun and Wu (Proc Am Math Soc 147(2):671–679, 2019). As a corollary, we show that for all 1 ? p? ? the expected Lp-approximation order of stochastic Bernstein polynomials is comparable to that given by the classical Bernstein polynomials.

Department(s)

Mathematics

Document Type

Article

DOI

10.1007/s00009-022-02150-y

Keywords

Dvoretzky–Kiefer–Wolfowitz inequality, modulus of continuity, order statistics, stochastic quasi-interpolation

Publication Date

10-1-2022

Journal Title

Mediterranean Journal of Mathematics

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