On probabilistic convergence rates of stochastic Bernstein polynomials

Abstract:

In this article, we introduce the notion “$L_p$-probabilistic convergence" ($1 \le p \le \infty$) of stochastic Bernstein polynomials built upon order statistics of identically, independently, and uniformly distributed random variables on $[0,1]$. We establish power and exponential convergence rates in terms of the modulus of continuity of a target function $f \in C[0,1]$. For $p$ in the range $1 \le p \le 2,$ we obtain Gaussian tail bounds for the corresponding probabilistic convergence. Our result for the case $p=\infty$ confirms a conjecture raised by the second and third authors. Monte Carlo simulations (presented at the end of the article) show that the stochastic Bernstein approximation scheme studied herein achieves comparable computational goals to the classical Bernstein approximation, and indicate strongly that the Gaussian tail bounds proved for $1 \le p \le 2$ also hold true for the cases $2< p \le \infty$.