Strong uniform laws of large numbers for bootstrap means and other randomly weighted sums

TL;DR

This paper proves new strong uniform laws of large numbers for a broad class of randomly weighted sums, including bootstrap and resampling methods, accommodating various weights and sample sizes.

Contribution

It extends existing strong laws to cover negatively orthant dependent weights and non-identically distributed weights, broadening the applicability of bootstrap methods.

Findings

Established strong uniform laws for bootstrap means.

Extended results to negatively orthant dependent weights.

Applicable to diverse resampling schemes and weighting procedures.

Abstract

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.