Decorrelation of Wavelet Coefficients for Long-Range Dependent Processes
Jan Mielniczuk, Piotr Wojdyłło · IEEE Transactions on Information Theory · 2007
We consider a discrete-time stationary long-range dependent process$(X_k)_{k\in Z}$such that its spectral density equals${\varphi}(\vert {\lambda}\vert)^{-2d}$, where${\varphi}$is a smooth function such that${\varphi}(0)={\varphi}^{\prime\prime}(0)=0$and${\varphi}({\lambda})\geq c{\lambda}$for${\lambda}\in [0,\pi]$. Then for any wavelet$\psi$with$N$vanishing moments, the lag$k$within-level covariance of wavelet coefficients decays as${\cal O}(k^{2d-2N-1})$when$k\to\infty$. The result applies to fractionally integrated autoregressive moving average (ARMA) processes as well as to fractional Gaussian noise.