Consistency of randomized integration methods

TL;DR

This paper proves the consistency of various randomized numerical integration methods, including $(t,d)$-sequences and Latin hypercube sampling, ensuring their estimators reliably converge to the true integral under certain conditions.

Contribution

It establishes the theoretical consistency of a broad class of randomized integration methods, including median modifications, for integrands in $L^p$ spaces.

Findings

Randomized methods are consistent in mean and probability.

Median modifications achieve almost sure convergence.

Applicability to integrands in $L^p$ with $p>1$.

Abstract

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 $L^p$ with $p>1$ consistency in terms of almost sure convergence