On the approximation of an integral by a sum of random variables (Q1270298)

It is investigated to estimate the integral of a smooth function on \([0,1]\) under the condition that only values of the function at \(n\) random points from the population uniformly distributed on \((0,1)\) can be observed. Two approximation methods are proposed; one is based on the trapezoidal rule and the other is based on Simpson's rule (generalized to the non-equidistant case). It is shown that the first approximation has an \(n^2\)-rate of convergence with a degenerate limiting distribution and the other approximation has the rate of convergence as fast as \(n^{3.5}\) with a Gaussian limiting distribution.