with a time varying $d$, namely, a stochastically evolving $d$ or a regime switching between, say, two values of $d$. These two approaches have been developed, respectively, in [@ray-tsay-2002] and [@haldrup-nielsen-2006] (see also [@guegan:lu:2009] where singularities in the spectral density at frequencies different from zero are considered in a piecewise stationary context).

As outlined earlier the alternative approach developed by Dahlhaus for locally stationary (short memory) processes is quite appealing as it allows a meaningful asymptotic study of the estimators. It was recently used for volatility estimation using time varying (short memory) non-linear processes, see [@dahlhaus-subbarao-2006; @fryzlewicz-sapatinas-subbarao-2008]. The first attempt to adapt Dahlhaus’s approach in the presence of long memory appears to be the unpublished preprint [@jensen:whitcher:2000]. The authors use the log linear relationship of the local variance of the maximum overlap discrete wavelet transform and their scaling parameter, plus a localization with a rectangular window in coefficient domain, to estimate the time-varying long memory parameter. However, the asymptotic analysis of the proposed estimator is not provided. Although it is not essential in their analysis, the considered model is a tvARFIMA$(q,d,r)$, see our examples below. An asymptotic analysis is provided in [@beran:2009] for a different estimator applied to the same 