Variance reduction order using good lattice points in Monte Carlo methods

Quasi-Monte Carlo methods and lattice rules with good lattice points give rapidly "good" approximations for numerical integration, but the error estimation is intractable in practice. In the literature, a randomization of these methods, using a combination of Monte Carlo and quasi-Monte Carlo methods, has been done to obtain a confidence interval using the Central Limit Theorem. In this paper we show that for a special class of functions with small Fourier coefficients and using good lattice points, the decreasing of the variance of the combined estimator is faster than the usual one.