Strong uniform laws of large numbers for bootstrap means and other randomly weighted sums. (English) Zbl 1543.60039
Summary: 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.
MSC:
| 60F15 | Strong limit theorems |
| 62G09 | Nonparametric statistical resampling methods |
| 62G05 | Nonparametric estimation |
Keywords:
empirical processes; generalized bootstrap; negatively orthant dependence; uniform Marcinkiewicz-Zygmund strong law; resamplingReferences:
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