A note on conditional versus joint unconditional weak convergence in bootstrap consistency results.

The consistency of a bootstrap or resampling scheme is classically validatedby weak convergence of conditional laws. However, when working with stochasticprocesses in the space of bounded functions and their weak convergence in theHoffmann-J{\o}rgensen sense, an obstacle occurs: due to possiblenon-measurability, neither laws nor conditional laws are well-defined. Startingfrom an equivalent formulation of weak convergence based on the boundedLipschitz metric, a classical circumvent is to formulate bootstrap consistencyin terms of the latter distance between what might be called a\emph{conditional law} of the (non-measurable) bootstrap process and the law ofthe limiting process. The main contribution of this note is to provide anequivalent formulation of bootstrap consistency in the space of boundedfunctions which is more intuitive and easy to work with. Essentially, theequivalent formulation consists of (unconditional) weak convergence of theoriginal process jointly with two bootstrap replicates. As a by-product, weprovide two equivalent formulations of bootstrap consistency for statisticstaking values in separable metric spaces: the first in terms of (unconditional)weak convergence of the statistic jointly with its bootstrap replicates, thesecond in terms of convergence in probability of the empirical distributionfunction of the bootstrap replicates. Finally, the asymptotic validity ofbootstrap-based confidence intervals and tests is briefly revisited, withparticular emphasis on the, in practice unavoidable, Monte Carlo approximationof conditional quantiles.