density at zero frequency which is parameterized by a time-varying long-memory parameter $d=d(u)$, $u\in[0,1]$, i.e. defined in rescaled time. Our approach is a true generalization of the stationary approach in that the latter corresponds to a time-constant $d$ for our locally stationary model. As in the case of [@moulines:roueff:taqqu:2007:jtsa], the long memory parameter is estimated by a log-regression of a series of wavelet scalograms (estimated wavelet variances per scale by summing the squared wavelet coefficients per scale over location) onto a range of scales (corresponding to the low frequency range of the spectrum). Although wavelets do not improve the estimation of $d$ in the standard stationary context $-1/2<d<1/2$, their use is of interest in various practical situations (presence of trends, under and over-differenced series leading to $d\geq1/2$ and $d\leq-1/2$ respectively), see details in [@fay-mouline-roueff-taqqu-2009]. However, in our work now the challenge is to *localize* the estimation of the no more constant parameter $d$. Wavelets are favorable in this situation since, in contrast to a Fourier analysis, they are well localized both over time and frequency, i.e. scale. The localization is achieved by smoothing over time the series of squared wavelet coefficients on each of the coarse scales which enter into 