On the approximation of an integral by a sum of random variables. (1998) - Einmahl, John H. J.; Van Zuijlen, Martien C. A.
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.