How Much Information Does Dependence Between Wavelet Coefficients Contain?

We focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic. To this end, we derive asymptotic distributional properties of the sample variance for a time series that is synthesized, ignoring some or all dependence between wavelet coefficients.

We show that the second-order properties differ from the those 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. In the case of sample autocovariances and sample autocorrelations at lag one, we indicate that first-order properties are erroneous.