  The block bootstrap approximates sampling distributions from dependent data
by resampling data blocks. A fundamental problem is establishing its
consistency for the distribution of a sample mean, as a prototypical statistic.
We use a structural relationship with subsampling to characterize the bootstrap
in a new and general manner. While subsampling and block bootstrap differ, the
block bootstrap distribution of a sample mean equals that of a $k$-fold
self-convolution of a subsampling distribution. Motivated by this, we provide
simple necessary and sufficient conditions for a convolved subsampling
estimator to produce a normal limit that matches the target of bootstrap
estimation. These conditions may be linked to consistency properties of an
original subsampling distribution, which are often obtainable under minimal
assumptions. Through several examples, the results are shown to validate the
block bootstrap for means under significantly weakened assumptions in many
existing (and some new) dependence settings, which also addresses a standing
conjecture of Politis, Romano and Wolf(1999). Beyond sample means, the
convolved subsampling estimator may not match the block bootstrap, but instead
provides a hybrid-resampling estimator of interest in its own right. For
general statistics with normal limits, results also establish the consistency
of convolved subsampling under minimal dependence conditions, including
non-stationarity.
