On the error distribution for randomly-shifted lattice rules

Randomized quasi-Monte Carlo (RQMC) methods estimate the expectation of a random variable by the average of n dependent realizations of it. In general, due to the strong dependence, the estimation error may not obey a central limit theorem. Analysis of RQMC methods have so far focused mostly on the convergence rates of asymptotic worst-case error bounds and variance bounds, when n ¿ ¿, but little is known about the limiting distribution of the error. Here we examine this limiting distribution for the special case of a randomly-shifted lattice rule, when the integrand is smooth. We start with simple one-dimensional functions, where we show that the limiting distribution is uniform over a bounded interval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. In higher dimensions, for linear functions, the distribution function of the properly standardized error converges to a spline of degree equal to the dimension.