On the approximation of an integral by a sum of random variables

[摘要] We approximate the integral of a smooth function on[0,1], where values are only known atnrandom points (i.e., a random sample from the uniform-(0,1)distribution), and at0and1. Our approximations are based on the trapezoidal rule and Simpson's rule (generalized to the non-equidistant case), respectively. In the first case, we obtain ann2-rate of convergence with a degenerate limiting distribution; in the second case, therate of con-vergence is as fast asn3½, whereas the limiting distribution isGaussian then.

[学科分类] 应用数学