  Existing frequency domain methods for bootstrapping time series have a
limited range. Consider for instance the class of spectral mean statistics
(also called integrated periodograms) which includes many important statistics
in time series analysis, such as sample autocovariances and autocorrelations
among other things. Essentially, such frequency domain bootstrap procedures
cover the case of linear time series with independent innovations, and some
even require the time series to be Gaussian. In this paper we propose a new,
frequency domain bootstrap method which is consistent for a much wider range of
stationary processes and can be applied to a large class of periodogram-based
statistics. It introduces a new concept of convolved periodograms of smaller
samples which uses pseudo periodograms of subsamples generated in a way that
correctly imitates the weak dependence structure of the periodogram. %The new
bootstrap procedure %corrects for those aspects of the distribution of spectral
means that cannot be mimicked by existing procedures. We show consistency for
this procedure for a general class of stationary time series, ranging clearly
beyond linear processes, and for general spectral means and ratio statistics.
Furthermore, and for the class of spectral means, we also show, how, using this
new approach, existing bootstrap methods, which replicate appropriately only
the dominant part of the distribution of interest, can be corrected. The finite
sample performance of the new bootstrap procedure is illustrated via
simulations.
