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A strong law of large numbers for scrambled net integration

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arxiv 2002.07859 v3 pith:HGMHKIVP submitted 2020-02-18 math.NA cs.NAmath.STstat.COstat.TH

classification math.NAcs.NAmath.STstat.COstat.TH
keywords stronglargenumbersintegrableintegrandintegrationnetsoptimization
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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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    math.NA 2019-08 conditional novelty 6.0 of 10

    RQMC estimation of CVaR sensitivity is strongly consistent and reaches mean error O(n^{-1/2-1/(4d-2)+epsilon}) under technical conditions.

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