On the approximation of an integral by a sum of random variables. (1998)

• Record Type: Journal Article
• Authors: Einmahl, John H. J.; Van Zuijlen, Martien C. A.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 11:Number 2(1998)
• Page Start: 107
• Page End: 114
• Publication Date: 1998
• Subjects: numerical integration -- order statistics -- spacings; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953398000100
• Languages: English
• ISSNs: 1048-9533

Abstract

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.