We prove that a class of randomized integration methods, including averages based on (t, d)-sequences, Latin hypercube sampling, Frolov points as well as Cranley-Patterson rotations, consistently estimates 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 Lp with p > 1 consistency in terms of almost sure convergence.(c) 2023 Elsevier Inc. All rights reserved.
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