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arXiv · 2002.07859

A strong law of large numbers for scrambled net integration

Art B. Owen·Daniel Rudolf
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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 $p$ for some $p>1$. 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 $L^2$ to $L^p$ for $p>1$ via the Riesz-Thorin interpolation theorem

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Art B. Owen, Daniel Rudolf. 2020-02-18. A strong law of large numbers for scrambled net integration. https://arxiv.org/abs/2002.07859

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