  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.
