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

Einmahl, John H. J.; Van Zuijlen, Martien C. A.

doi:https://doi.org/10.1155/S1048953398000100

International Journal of Stochastic Analysis

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Volume 11 |Article ID 670648 | https://doi.org/10.1155/S1048953398000100

John H. J. Einmahl, Martien C. A. Van Zuijlen, "On the approximation of an integral by a sum of random variables", International Journal of Stochastic Analysis, vol. 11, Article ID 670648, 8 pages, 1998. https://doi.org/10.1155/S1048953398000100

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, The Netherlands

²University of Nijmegen, Department of Mathematics, Toernooiveld 1, The Netherlands

Received01 Oct 1996

Revised01 Sep 1997

Abstract

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.

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Copyright © 1998 Hindawi Publishing Corporation. 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.

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