Random Riemann Sum estimator versus Monte Carlo

Two estimators of the expectation of a function, the classical based in Monte Carlo sampling method and one based in Random Riemann Sums, are compared. It presents the differences on bias, variance, convergence and mainly convergence rates. Two ways of sampling to obtain a Random Riemann Sum estimator are given. The first one provides a sequence of estimations whose terms are independent, this fact produces a loss of order one in the convergence rate for the strong law compared with Monte Carlo sampling method. The second one is considered in order to improve these results.