Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Variance Reduction via Lattice Rules
Management Science
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Numerical Methods in Scientific Computing: Volume 1
Numerical Methods in Scientific Computing: Volume 1
A practical view of randomized quasi-Monte Carlo: invited presentation, extended abstract
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
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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.