assumption $M\geq p$ because in this case the wavelet coefficients at given scale are weakly stationary. Moreover, by (\[eq:genLocSpecDens\]) and (\[eq:HjTilde\]), $$\sigma^2_j(u)  = \int_{-\pi}^\pi \left|{H}_j(\lambda)\right|^2\,f(u;\lambda)\;\rmd \lambda 
= \int_{-\pi}^\pi \left|\tilde{H}_j(\lambda)\ A(u;\lambda)\right|^2\;\rmd \lambda \;.$$ The following intuitive definition will be also useful when developing our estimation theory using stationary approximations. For any $u\in [0,1]$ one may define a *tangent* stationary process for the $p$-th increment $$\label{eq:tangentProcSpectral}
\diffop^pX_{t}(u) = \int_{-\pi}^\pi A(u;\lambda)\ \rme^{\rmi\lambda t}\;\rmd Z(\lambda)\ ,$$ whose spectral density is $|1- \rme^{-\rmi\lambda}|^{2p}f(u,\lambda)$. Further we define the wavelet coefficients of the tangent weakly stationary process at any $u\in [0,1]$, namely, $$\begin{aligned}
  \label{eq:wavcoeffStat}
  W_{j,k}(u) &= \sum_{t=1}^T \tilde{h}_{j,2^jk-t}(\diffop^p X)_{t}(u)\\
  \label{eq:wavcoeffStatSpectral}
&=\int_{-\pi}^\pi \tilde{H}_j(\lambda)\ A(u;\lambda)\ \rme^{\rmi\lambda 2^jk}\;\rmd Z(\lambda)  \;,
\quad k=0,\dots,T_j-1\;.\end{aligned}$$ these wavelet coefficients are indeed those of a process with generalized spectral density $f(u,\cdot)$. Thus the above definition gives $$\label{eq:defLocWavSpec}
\sigma^2_j(u)= \E\left[W_{j,k}^2(u)\right]  \;.$$

0.3cm

An important tool for the estimation of the long memory is the *scalogram* (first introduced in this context by [@wornell:oppenheim:1992] and developed by [@abry:veitch:1998]) defined as $$\widehat{\sigma}^2_{j}=T_j^{-1}\sum_{k=0}^{T_j-1}W_{j,k}^2 \; .$$ Here to cope with local stationarity, we will need a *local scalogram* for estimating the local wavelet spectrum, namely, for a given $u\in [0,1]$, $$\label{eq:EstimatorWeights}
\widehat{\sigma}^2_{j,T}(u)= \sum_{k=0}^{T_j-1}\gamma_{j,T}(k)W_{j,k;T}^2 \; ,$$ where $\{\gamma_{j,T}(k)\}$ are some non-negative weights 