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

John Einmahl and Martien C. A. Van Zuijlen

International Journal of Stochastic Analysis, 1998, vol. 11, 1-8

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 n 2 -rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as n 3 ½ , whereas the limiting distribution is Gaussian then.

Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:670648

DOI: 10.1155/S1048953398000100

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