the regression and $L$ is the lowest scale index used in the regression. Let $\bw$ be a vector $\bw =[w_0,\dots,w_{\ell}]^T$ satisfying $$\label{eq:propertyw}
\sum_{i=0}^{\ell} w_{i}  = 0\quad\text{and}\quad 2 \log(2) \sum_{i=0}^{\ell} i w_{i}  = 1 \;.$$ The local estimator of $d(u)$ is defined as $$\label{eq:definition:estimator:regression}
\hat{d}_T(L) = \sum_{j=L}^{L+\ell} w_{j-L} \log \left( \widehat{\sigma}^2_{j,T}(u)\right) \; .$$

Conditions on the weights $\gamma_{j,T}(k)$ and examples {#sec:exampl-weights}
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Let us now precise our conditions on the weights $\gamma_{j,T}(k)$. Denote, for any $0\leq i\leq j$, $v\in\{0,\dots,2^{i}-1\}$ and $\lambda\in\R$, $$\label{eq:defweightsFourier}
\Phi_{j,T}(\lambda;i,v)=\sum_{l\in \calT_j(i,v)}\gamma_{j-i,T}(2^i l+v)\rme^{\rmi l \lambda}\;,$$ where $$\label{eq:calT}
   \calT_j(i,v)= \left\{l \,:\, 0\leq l < 2^{-i}(T_{j-i}-v)\right\}\;.$$ We moreover define $$\label{eq:deltaDef}
\delta_{j,T}=\sup_{k=0,\dots,T_j-1}|\gamma_{j,T}(k)| \; .$$

The weights $\gamma_{j,T}(k)$ must satisfy an appropriate asymptotic behavior as $T\to\infty$ for obtaining a consistent estimator of $d(u)$. More precisely, the following assumption will be required.

\[assump:weights\] The index $j$ depends on $T$ so that the weights $(\gamma_{j,T}(k))_{k}$ satisfy the following asymptotic properties as $T\to\infty$.

(i) \[item:delatAssump\] We have $\delta_{j,T}\to0$, and for any fixed integer $i$, $\delta_{j+i,T}\sim2^{i}\delta_{j,T}$.

(ii) \[item:Vassump\] For all $i,i'\geq0$, $v\in\{0,\dots,2^{i}-1\}$ and $v'\in\{0,\dots,2^{i'}-1\}$, there exists a constant $V=V(i,v;i',v')$ such that $$\label{eq:VasympDef}
    \delta_{j,T}^{-1}\int_{-\pi}^\pi \Phi_{j,T}(\lambda;i,v)\overline{\Phi_{j,T}(\lambda;i',v')}\,\rmd\lambda \to V(i,v;i',v')\;.$$

(iii) \[item:neglicAssump\] For all $\eta>0$, $i\geq0$ and $v\in\{0,\dots,2^{i}-1\}$, we have $$\delta_{j,T}^{-1/2} \sup_{\eta\leq|\lambda|\leq\pi}\left|\Phi_{j,T}(\lambda;i,v)\right|\to 0 \; .$$

(iv) \[item:Locassump\] For $q=0,1,2$, we have 