On the approximation of an integral by a sum of random variables
We approximate the integral of a smooth function on , where values are only known at random points (i.e., a random sample from the uniform- distribution), and at and . 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 -rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as , whereas the limiting distribution is Gaussian then.
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