How much information does dependence between wavelet coefficients contain?

Motivated by several papers which propose statistical inference assuming independence of wavelet coefficients for both short- as well as long-range dependent time series, we focus exemplary on the sample variance and investigate the influence of the dependence between wavelet coefficients to this statistic. To this end, we derive asymptotic distributional properties of the sample variance for a time series which is synthesized ignoring some or all dependence between wavelet coefficients. We show that the second order properties differ from the ones of the true time series whose wavelet coefficients have the same marginal distribution except in the independent Gaussian case. This holds true even if the dependency is correct within each level and only the dependence between levels is ignored. For the example of sample autocovariances and sample autocorrelations at lag one, we indicate that already first order properties are erroneous in these cases. In a second step, several non-parametric bootstrap schemes in the wavelet domain are investigated which take more and more dependence into account until finally the full dependency is mimicked. We obtain very similar results, where only a bootstrap mimicking the full covariance structure correctly can be valid asymptotically. A simulation study supports our theoretical findings for the wavelet domain bootstraps. For long-range dependent time series with long-memory parameter <i>d</i> &gt; 1/4, we show that some additional problems occur which cannot be solved easily without using additional information for the bootstrap.