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A strong law of large numbers for scrambled net integration

Abstract

This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integral over others, arising in Bayesian optimization. This strong law requires that the integrand have a finite moment of order pp for some p>1p>1. Previously known results implied a strong law only for Riemann integrable functions. Previous general weak laws of large numbers for scrambled nets require a square integrable integrand. We generalize from L2L^2 to LpL^p for p>1p>1 via the Riesz-Thorin interpolation theorem

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