$$\label{eq:NormalizedWeightsSums}
      \Gamma_q(u;j,T)=O\left((\delta_{j,T})^{-q}\right)\;,$$ where $\Gamma_q(u;j,T)$ is defined in (\[eq:GammaDefQ\]).

We provide several examples of weights satisfying this assumption below. In these examples, the weights $\gamma_{j,T}(k)$, $k=0,\ldots,T_j$, are entirely determined by $T_j$ and a bandwidth parameter $\band_T$ and $$\label{eq:bandwidthDef}
\delta_{j,T}^{-1}\asymp \band_T T_j \sim \band_T T2^{-j} \;.$$ In kernel estimation, one may interpret the bandwidth parameter $\band_T$ as the proportion of wavelet coefficients used for the estimation of the local scalogram $\widehat{\sigma}^2_{j,T}(u)$ at given scale $j$ and position $u$, among the $T_j$ wavelet coefficients available at scale $j$ from $T$ observations $X_{1,T},\dots,X_{T,T}$. Lemmas \[lem:two-sided-kernel\] and \[lem:recursive-weights\] show that, for these examples, Assumption \[assump:weights\] is satisfied as soon as $T_j\to\infty$ and $\band_T T_j\to0$, except in the non-compactly supported case (K-\[item:exp\]) in Lemma \[lem:two-sided-kernel\] where we assume in addition that $T_j\exp(-c' \band_{T}T_j)=O(1)$ for any $c'>0$, which holds in the typical asymptotic setting $\band_T \asymp T_j^{-\zeta}$ with $\zeta\in(0,1)$.

\[ex:two-sided-kernel\] For $u\in(0,1)$, one has a number of observations before rescaled time $u$ and after rescaled time $u$ both tending to infinity. Thus we may use a two-sided kernel to localize the memory parameter estimator around $u$. For a given bandwidth $b=b_T$, the corresponding weights are given by $$\label{eq:KernelWeights}
\gamma_{j,T}(k)=\rho_{j,T}^{-1}\;K( (uT_j-k) / (\band_{T}T_j) )\; ,$$ where $K$ is 