derive asymptotic results for $\widehat{\sigma}^2_{j,T}(u)$, we first establish a bound on the error made when approximating $\widehat{\sigma}^2_{j,T}(u)$ by $\widetilde{\sigma}^2_{j,T}(u)$.

\[prop:local-scalogram\] Let $u\in[0,1]$ and consider a model satisfying Assumption \[ass:model\_lin\]. Assume \[item:Wreg\]–\[item:MIM\] hold with $M\geq
  p\vee (d(u)-1/2)$ and $\alpha>1/2-d(u)$. Suppose moreover that Assumption \[assump:weights\](\[item:Locassump\]) hold. Then, the following approximation holds. $$\label{eq:ScalApproxResult}
    \E\left[\left(\widehat{\sigma}^2_{j,T}(u)-\widetilde{\sigma}^2_{j,T}(u)\right)^2\right]=
 O\left(2^{(6+4p)j}T^{-4} \delta_{j,T}^{-4} + 2^{(3+2p+2d(u))j}T^{-2} \delta_{j,T}^{-2}\right) \; .$$

Next, we derive a bound of the mean square error for estimating $f^*(u,0)\constantePsiD(d(u))\;2^{2jd(u)}$ using the estimator $\widehat{\sigma}^2_{j,T}(u)$, where $\constantePsiD$ is the function defined by $$\label{eq:Kdef}
  \constantePsiD(d)= \int_{-\infty}^{\infty}|\xi|^{-2d}\,|\hat\psi(\xi)|^2\;\rmd\xi\;,\quad   1/2-\alpha<d< M+1/2\;.$$ In fact as the estimator $\hat{d}_T(L)$ is defined in (\[eq:definition:estimator:regression\]) using $\widehat{\sigma}^2_{j,T}(u)$ with $j=L+i$, $i=0,\dots,\ell$, and as $L,T\to\infty$, it will be convenient to normalize these quantities by $2^{2Ld(u)}$, so that $f^*(u,0)\constantePsiD(d(u))\;2^{2jd(u)}/2^{2Ld(u)}=f^*(u,0)\constantePsiD(d(u))\;2^{2id(u)}$ does not depend on $L$.

\[theo:rate-param-scalo\] Let $u\in[0,1]$ and consider a model satisfying Assumption \[ass:model\_lin\]. Assume \[item:Wreg\]–\[item:MIM\] hold with $M\geq p\vee d(u)$ and $\alpha>(1+\beta)/2 -d(u)$. Then we have, as $j\to\infty$, $$\label{eq:BiasControl}
\sigma_j^2(u)=
f^*(u,0)\constantePsiD(d(u))\;2^{2jd(u)}\left\{1+O\left(2^{-\beta j}\right)\right\}\;.$$ Suppose moreover that Assumption \[assump:weights\] holds and that $$\label{eq:condConsistency}
    2^{(3+2\{p-d(u)\})L}T^{-2} \delta_{L,T}^{-2}\to0\;.$$ Then we have for $j=L+i$ with $i=0,\dots,\ell$, $$\begin{gathered}
\label{eq:ratesOp}
\E\left[(2^{-2Ld(u)}\widehat{\sigma}^2_{j,T}(u)-f^*(u,0)\constantePsiD(d(u))\;2^{2id(u)})^2\right]\\
=O\left(\delta_{L,T}+2^{(3+2\{p-d(u)\})L}T^{-2} \delta_{L,T}^{-2}+2^{-2\beta L}\right)\end{gathered}$$

Using the approximation result stated in Proposition \[prop:local-scalogram\], we may also wish to obtain a central limit theorem (CLT) for the local scalogram. To 