Asymptotic Decorrelation of Wavelet Packet Transform for Certain Long-Memory Processes

Abstract

The asymptotic decorrelation of the discrete wavelet transform for long-memory processes (such as the fractionally differenced (FD) process, the autoregressive fractionally integrated moving average (ARFIMA) process, and the Gegenbauer autoregressive moving average (GARMA) process) as well as the statistical inference techniques based on this property, have received much attention nowadays. In this paper, we investigate the asymptotic decorrelation property of the discrete wavelet packet transform (DWPT) for two classes of discrete-time long-memory processes containing the ARFIMA and the GARMA processes. Especially, we prove theoretically that the covariance across between-packet DWPT coefficients decays hyperbolically or exponentially fast as the width of the underlying Daubechies scaling and wavelet filters to generate the DWPT gets large. Meanwhile, we show that the covariance between within-packet DWPT coefficients converges hyperbolically fast to its corresponding counterpart when the underlying scaling and wavelet filters to generate the DWPT are the Shannon's ideal low- and high-pass filters.