  We consider a time series $X=\{X_k, k\in\mathbb{Z}\}$ with memory parameter
$d\in\mathbb{R}$. 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 $d$. 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.
