  Because the stationary bootstrap resamples data blocks of random length, this
method has been thought to have the largest asymptotic variance among block
bootstraps Lahiri [Ann. Statist. 27 (1999) 386--404]. It is shown here that the
variance of the stationary bootstrap surprisingly matches that of a block
bootstrap based on nonrandom, nonoverlapping blocks. This argument translates
the variance expansion into the frequency domain and provides a unified way of
determining variances for other block bootstraps. Some previous results on the
stationary bootstrap, related to asymptotic relative efficiency and optimal
block size, are also updated.
