the log-regression, giving raise to a *local scalogram*. We propose both a more traditional method based on two-sided kernels and also a recursive scheme of one-sided smoothing weights, adapted to the end point of the observation period. The model studied in this paper arises naturally in the now long history of time series modelling in presence of persistent memory. A survey on this subject is provided in [@samo:2006]. In Chapter 3 of this reference, it is recalled how long memory and non stationarity have been used as concurrent modelling approaches, in particular for financial data, see *e.g.* [@mikosch-starica-2004]. Long memory modeling for financial data goes back to [@ding-granger-engle1993]. Originally investigated on absolute powers of stock returns, long memory models are currently widely used for realized volatility data since they were proven efficient for forecasting purposes in this context in [@ABDL-2003]. It is interesting to note that only 3 years after [@ding-granger-engle1993], the need for time varying long memory parameters was pointed out in [@ding-granger-1996], see Section 5 and in particular Figure 6 where an estimated $d$ is plotted evolving between the values $0.358$ and $0.714$ over a 60 years long period. In this reference two approaches are suggested for coping 