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

J.H.J. Einmahl, M.C.A. Zuijlen, van

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    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.
    Original languageEnglish
    Pages (from-to)107-114
    JournalJournal of Applied Mathematics and Stochastic Analysis
    Volume11
    Issue number2
    DOIs
    Publication statusPublished - 1998

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