Title: A note on conditional versus joint unconditional weak convergence in bootstrap consistency results,
Abstract: The consistency of a bootstrap or resampling scheme is classically validated
by weak convergence of conditional laws. However, when working with stochastic
processes in the space of bounded functions and their weak convergence in the
Hoffmann-J{\o}rgensen sense, an obstacle occurs: due to possible
non-measurability, neither laws nor conditional laws are well-defined. Starting
from an equivalent formulation of weak convergence based on the bounded
Lipschitz metric, a classical circumvent is to formulate bootstrap consistency
in terms of the latter distance between what might be called a
\emph{conditional law} of the (non-measurable) bootstrap process and the law of
the limiting process. The main contribution of this note is to provide an
equivalent formulation of bootstrap consistency in the space of bounded
functions which is more intuitive and easy to work with. Essentially, the
equivalent formulation consists of (unconditional) weak convergence of the
original process jointly with two bootstrap replicates. As a by-product, we
provide two equivalent formulations of bootstrap consistency for statistics
taking values in separable metric spaces: the first in terms of (unconditional)
weak convergence of the statistic jointly with its bootstrap replicates, the
second in terms of convergence in probability of the empirical distribution
function of the bootstrap replicates. Finally, the asymptotic validity of
bootstrap-based confidence intervals and tests is briefly revisited, with
particular emphasis on the, in practice unavoidable, Monte Carlo approximation
of conditional quantiles.