# Convolved subsampling estimation with applications to block bootstrap

## TL;DR

This paper introduces a new perspective on block bootstrap using convolved subsampling, providing simple conditions for its consistency and validating its effectiveness under minimal assumptions for dependent data.

## Contribution

It establishes a general framework linking subsampling and block bootstrap, offering necessary and sufficient conditions for normal limit consistency in dependent data settings.

## Key findings

- Validates block bootstrap for means under weakened dependence assumptions
- Provides conditions for convolved subsampling to achieve normal limits
- Addresses a longstanding conjecture on bootstrap consistency

## Abstract

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.