SIAM Review, Volume 63, Issue 2, Page 360-372, January 2021.
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.

中文翻译：

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用于加扰网络集成的强大数定律

SIAM 评论，第 63 卷，第 2 期，第 360-372 页，2021 年 1 月。
本文提供了强大的大数定律，用于在通过嵌套统一打乱随机化的数字网络上进行集成。激励问题是优化贝叶斯优化中出现的积分变量的某些变量。这个强定律要求被积函数对于某些 $p>1$ 有一个有限阶矩 $p$。先前已知的结果暗示了仅适用于黎曼可积函数的强定律。以前加扰网络的大数一般弱定律需要平方可积被积函数。我们通过 Riesz--Thorin 插值定理从 $L^2$ 推广到 $L^p$，因为 $p>1$。