pay on the rate of estimation (although we are unable to prove that this price is optimal). It is not as surprising as it may appear. To understand why, consider a piecewise stationary context where a finite number of regime switching times occur over the observation sample. One clearly sees that the long memory over each stationary segment can be estimated at the same rate as in the stationary context. As we will see, the picture is more complicated in a locally stationary context but it can still be handled. More precisely, we will show that looking at low frequencies is allowed in a locally stationary model but with an additional cost on the rate of convergence depending on how small the frequency used for the analysis is. We believe that such theoretical results are crucial for the practical estimation of the time varying long memory parameter as they demonstrate the viability of such an approach while indicating that it should be applied with care.

Summarizing our results, the rest of the paper is organized as follows. In Section \[sec:2\], we give the technical details of our locally stationary long memory model of semi-parametric type and give a series of 