P. F. Craigmile, P. Guttorp and D. B. Percival (2004), `Trend Assessment in a Long Memory Dependence Model using the Discrete Wavelet Transform,' Environmetrics, 15(4), pp. 313-35.
In recent years there has been much interest in the analysis of time series using a discrete wavelet transform (DWT) based upon a Daubechies (1992) wavelet filter. Part of this interest has been sparked by the fact that the DWT approximately decorrelates certain stochastic processes, including stationary fractionally differenced (FD) processes with long memory characteristics and certain nonstationary processes such as fractional Brownian motion. In this paper we show that, as the width of the wavelet filter used to form the DWT increases, the covariance between wavelet coefficients associated with different scales decreases to zero for a wide class of stochastic processes. These processes are Gaussian with a spectral density function (SDF) that is the product of the SDF for a (not necessarily stationary) FD process multiplied by any bounded function that can serve as an SDF on its own. We demonstrate that this asymptotic theory provides a reasonable approximation to the between-scale covariance properties of wavelet coefficients based upon filter widths in common use. Our main result is one important piece of an overall strategy for establishing asymptotic results for certain wavelet-based statistics.
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