Integration of bounded monotone functions: Revisiting the nonsequential case, with a focus on unbiased Monte Carlo (randomized) methods

In this article we revisit the problem of numerical integration for monotone bounded functions, with a focus on the class of nonsequential Monte Carlo methods. We first provide new a lower bound on the maximal $L^p$ error of nonsequential algorithms, improving upon a theorem of Novak when p > 1. Then we concentrate on the case p = 2 and study the maximal error of two unbiased methods-namely, a method based on the control variate technique, and the stratified sampling method.