A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series

We consider a time series X = (X-k, k is an element of Z) with memory parameter d(0) is an element of R. This time series is either stationary or can be made stationary after differencing a finite number of times. We study the "local Whittle wavelet estimator" of the memory parameter d(0). This is a wavelet-based semiparametric pseudo-likelihood maximum method estimator. The estimator may depend on a given finite range of scales or on a range which becomes infinite with the sample size. We show that the estimator is consistent and rate optimal if X is a linear process, and is asymptotically normal if X is Gaussian.