localized at indices $k\approx uT_j$ and normalized in such a way that $$\label{eq:normWeights}
\sum_{k=0}^{T_j-1}\gamma_{j,T}(k) =1 \; .$$ The localization property will be expressed by imposing a bound on the increase rate of the following quantity (see equation (\[eq:NormalizedWeightsSums\])) $$\label{eq:GammaDefQ}
\Gamma_q(u;j,T)=\sum_{k=0}^{T_j-1}|\gamma_{j,T}(k)|\;\;|k-Tu2^{-j}|^q  \;,$$ as $T\to\infty$ for appropriate values of the exponent $q$.

An important tool for studying the local scalogram is the *tangent scalogram* defined as $$\label{eq:tangentScalogram}
\widetilde{\sigma}^2_{j,T}(u)=\sum_{k=0}^{T_j-1}\gamma_{j,T}(k)W_{j,k}^2(u)\;.$$ We note that this definition is similar to that of the local scalogram in (\[eq:EstimatorWeights\]) but with the wavelet coefficients $W_{j,k;T}$ replaced by the tangent wavelet coefficients $W_{j,k}^2(u)$ defined in (\[eq:wavcoeffStat\]). The tangent scalogram is not an estimator since it cannot be computed from the observations $X_{1,T},\dots,X_{T,T}$. However, it provides useful approximations of the local scalogram.

We conclude this section by deriving an [*estimator of the time-varying long memory parameter*]{}. The local wavelet spectrum is related to the local memory parameter $d(u)$ by the asymptotic property $\sigma^2_j(u)\sim c
2^{2d(u)j}$ as $j\to\infty$. This property will be made more precise when we study the bias, see the relation (\[eq:BiasControl\]) below. An estimator of $d(u)$ is obtained by a linear regression of $(\log \widehat{\sigma}^2_{j,T}(u))_{j=L,\dots,L+\ell}$ with respect to $j=L,\dots,L+\ell$, where $\ell$ is the number of scales used in 