Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 2, Pages 107-114

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

John H. J. Einmahl1 and Martien C. A. Van Zuijlen2

1Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, Eindhoven 5600 MB, The Netherlands
2University of Nijmegen, Department of Mathematics, Toernooiveld 1, Nijmegen 6525 ED, The Netherlands

Received 1 October 1996; Revised 1 September 1997

Copyright © 1998 John H. J. Einmahl and Martien C. A. Van Zuijlen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We approximate the integral of a smooth function on [0,1], where values are only known at n random points (i.e., a random sample from the uniform-(0,1) distribution), and at 0 and 1. 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 an n2-rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as n3½, whereas the limiting distribution is Gaussian then.