On the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes

In this correspondence, we develop an asymptotic theory for the correlations of wavelet coefficients of the discrete wavelet transform (DWT) for fractionally differenced processes. It provides a theoretical justification for the approximate decorrelation property of the DWT for fractionally differenced processes. In addition, it provides insights on how the length of the wavelet filter affects the within scale correlations and the between scale correlations differently; for within scale correlations, increasing the length of the wavelet filter increases the rate of decay as the two wavelet coefficients get further apart, while for between scale correlations, using a wavelet filter that is long enough can reduce the between scale correlations even for wavelet coefficients that are close together.