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The block bootstrap approximates sampling distributions from dependent databy resampling data blocks. A fundamental problem is establishing itsconsistency for the distribution of a sample mean, as a prototypical statistic.We use a structural relationship with subsampling to characterize the bootstrapin a new and general manner. While subsampling and block bootstrap differ, theblock bootstrap distribution of a sample mean equals that of a $k$-foldself-convolution of a subsampling distribution. Motivated by this, we providesimple necessary and sufficient conditions for a convolved subsamplingestimator to produce a normal limit that matches the target of bootstrapestimation. These conditions may be linked to consistency properties of anoriginal subsampling distribution, which are often obtainable under minimalassumptions. Through several examples, the results are shown to validate theblock bootstrap for means under significantly weakened assumptions in manyexisting (and some new) dependence settings, which also addresses a standingconjecture of Politis, Romano and Wolf(1999). Beyond sample means, theconvolved subsampling estimator may not match the block bootstrap, but insteadprovides a hybrid-resampling estimator of interest in its own right. Forgeneral statistics with normal limits, results also establish the consistencyof convolved subsampling under minimal dependence conditions, includingnon-stationarity.

Convolved subsampling estimation with applications to block bootstrap