Consistency of randomized integration methods

We prove that a class of Monte Carlo methods, including averages based on randomized digital nets, Latin hypercube sampling, randomized Frolov as well as Cranley-Patterson rotated point sets, consistently estimate expectations of integrable functions. Consistency here refers to convergence in mean and/or convergence in probability of the estimator to the integral of interest. Moreover, we suggest median modified methods and show for integrands in $L^p$ with $p>1$ consistency in terms of almost sure convergence.Comment: 17 pages. The main results have been improved and an error has been correcte