Effects of block lengths on the validity of block resampling methods

Resampling methods for dependent variables typically involve selecting blocks of observations. In this paper, we investigate the effects of different block lengths on the consistency of approximations generated by block bootstrap and subsampling methods. A complete description is provided for the case of sample mean. It is shown that both methods are consistent if the block length grows at a rate slower than the sample size. When the growth rate of block lengths is comparable to the sample size, the resulting approximations are no longer consistent. In the latter case, we present a detailed account of the limit behavior of these resampling distribution approximations, considered as random elements in the space of all probability measures on the real line.