Stochastic Quasi-Interpolation with Bernstein Polynomials

We introduce the notion “stochastic quasi-interpolation on compact Hausdorff spaces”, and establish Gaussian-type L^p -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 L^p -approximation order of stochastic Bernstein polynomials is comparable to that given by the classical Bernstein polynomials.