This article establishes novel strong uniform laws of large numbers for randomly weighted sums such as bootstrap means. By leveraging recent advances, these results extend previous work in their general applicability to a wide range of weighting procedures and in their flexibility with respect to the effective bootstrap sample size. In addition to the standard multinomial bootstrap and the m-out-of-n bootstrap, our results apply to a large class of randomly weighted sums involving negatively orthant dependent (NOD) weights, including the Bayesian bootstrap, jackknife, resampling without replacement, simple random sampling with over-replacement, independent weights, and multivariate Gaussian weighting schemes. Weights are permitted to be non-identically distributed and possibly even negative. Our proof technique is based on extending a proof of the i.i.d. strong uniform law of large numbers to employ strong laws for randomly weighted sums; in particular, we exploit a recent Marcinkiewicz--Zygmund strong law for NOD weighted sums.

翻译：本文建立了关于随机加权和（如Bootstrap均值）的新型强大数律。通过利用最新进展，这些结果在权重过程的广泛适用性及有效Bootstrap样本量的灵活性方面拓展了以往工作。除标准多项Bootstrap和m-out-of-n Bootstrap外，我们的结果适用于包含负象限相依（NOD）权重的一大类随机加权和，包括贝叶斯Bootstrap、刀切法、无放回重抽样、带超代换的简单随机抽样、独立权重及多元高斯加权方案。权重允许非相同分布甚至可能为负值。我们的证明技术基于拓展独立同分布强大数律的证明方法，以适用于随机加权和的强大数律；特别地，我们利用了近期针对NOD加权和的Marcinkiewicz-Zygmund强大数律。