mainly restricted to the short-range dependent case. One prominent approach, that we adopt in this paper, too, is the model of local stationarity, introduced by a series of papers by R. Dahlhaus ([@dahlhaus96; @dahlhaus:1997; @dahlhaus2000]): in a non-parametric set-up, the spectral structure of the underlying stochastic process is allowed to be smoothly varying over time. Of course, time-varying linear processes (of ARMA type) arise as a subclass of these locally stationary processes. In order to come up with a rigorous asymptotic theory of consistency and inference, the time-dependence of the spectral density $f(u,\lambda)$ of such locally stationary processes is modelled to be in rescaled time $u\ \in
[0,1]$, leading to a problem of non-parametric curve estimation: increasing the sample size $T$ of the observed time series does no more mean to look into the future but to dispose of more and more observations to identify $f(t/T,\lambda)$ locally around the “reference” rescaled time point $u
\approx t/T$.

In the aforementioned spirit of semi-parametric modelling, and in contrast to the parametric approach of [@beran:2009], one of the very few existing approaches on time-varying long-memory modelling, we consider in this paper a locally stationary long-range dependent process with a singularity in the spectral 