exponent* corresponding to the *bandwidth parameter* $\band_{T}$. For any $u\in (0,1]$, we define a local recursive scalogram by $$\widehat{\sigma}^2_{j;T}(u)=\rho_{j,T}^{-1} \widehat{\sigma}^2_{j,[u\,T_j]-1;T}\; ,$$ where $[a]$ denotes the integer part of $a$ and $$\label{eq:rec_rho}
\rho_{j,T}=\sum_{k=0}^{[u\,T_j]-1} \rme^{-k/(\band_{T}T_j)}
= \frac{1-\rme^{-[u\,T_j]/(\band_{T}T_j)}}{1-\rme^{-(\band_{T}T_j)^{-1}}} \;.$$ Hence (\[eq:EstimatorWeights\]) and (\[eq:normWeights\]) hold with $$\label{eq:recWeights}
  \gamma_{j,T}(k)= \rho_{j,T}^{-1} \rme^{-([u\,T_j]-1-k)/(\band_{T}T_j)} \1_{[0,uT_j-1)}(k) \;.$$ Observe that these weights are one-sided by construction, since only the observations before rescaled time $u$ are used for estimating $d(u)$. This is the reason why we may take $u\in (0,1]$. Lemma \[lem:recursive-weights\] shows that Assumption \[assump:weights\] holds for these weights, provided that $\band_{T}\to0$ and $T_j\band_{T}\to\infty$.

Asymptotic properties {#sec:4}
=====================

We study the asymptotic properties of $\widehat{d}_T(L)$ defined by (\[eq:definition:estimator:regression\]) as $L,T\to\infty$ in such a way that Assumption \[assump:weights\] holds for each $j=L,L+1,\dots,L+\ell$ and for the chosen weights $\gamma_{j,T}(k)$. We provide further conditions on $L,T,\delta_{L,T}$ under which consistency holds and derive the corresponding rate of convergence (Corollary \[cor:rate-param-est\]). Under strengthened conditions, we further show that $\widehat{d}_T(L)$ is asymptotically normal (Theorem \[theo:clt-param-est\]). These results essentially follow from asymptotic results on the tangent scalogram (Theorem \[thm:clt-tang-scal\], Relations (\[eq:BiasControl\]) and (\[eq:VarTildeBound\])) and approximation results on the local scalogram (Proposition \[prop:local-scalogram\]) based on the tangent scalogram.

Asymptotic properties of the local scalogram
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In order to 