TY - JOUR

T1 - On the approximation of an integral by a sum of random variables

AU - Einmahl, J.H.J.

AU - Zuijlen, van, M.C.A.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

U2 - 10.1155/S1048953398000100

DO - 10.1155/S1048953398000100

M3 - Article

SN - 1048-9533

VL - 11

SP - 107

EP - 114

JO - Journal of Applied Mathematics and Stochastic Analysis

JF - Journal of Applied Mathematics and Stochastic Analysis

IS - 2

ER -