based on wavelet analysis {#sec:3}
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Discrete wavelet transform (DWT)
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Following the approach presented in [@moulines:roueff:taqqu:2007:jtsa] for the estimation of the memory parameter of a stationary sequence, we compute the discrete wavelet transform (DWT) of $\{X_{t,T},\;1\leq t \leq T\}$ (in discrete time) for a given scale function $\phi$ and wavelet $\psi$. We denote by $\{W_{j,k;T}\,;j\geq0,k\in\mathbb{Z}\}$ the wavelet coefficients of the process $\{X_{t,T},\,1\leq t\leq T\}$, $$\label{eq:wavcoeffLocStat}
  W_{j,k;T}= \sum_{t=1}^T h_{j,2^jk-t}X_{t,T},\quad k=0,\dots,T_j-1\;,$$ where $\{h_{j,t},\;,t\in\Z\}$ denotes the wavelet detail filter at scale $j$ associated to $\phi$ and $\psi$ through the relation $$h_{j,t}=2^{-j/2}\int_{-\infty}^\infty\phi(u+t)\psi(2^{-j}u)\;\rmd u\;,$$ and $T_j$ the number of available wavelet coefficients at scale $j$, which satisfies $$\label{eq:Tjasymp}
T2^{-j} - c \leq T_j\leq T2^{-j},\quad\text{for some constant $c$ independent of $j\geq 0$}\;.$$ The filter $h_{j,\cdot}$ and $T_j$ are defined so that the support $\{t~:~h_{j,2^jk-t}\neq0\}$ is included in $\{1,\dots T\}$ for $k=0,\dots,T_j-1$. Observe that here $j$ denotes the scale index of the wavelet coefficient and $k$ its position index. We use the convention that a large $j$ corresponds to a coarse scale. Let us define $$\label{eq:HjDef}
H_j(\lambda)=\sum_{t\in\Z} h_{j,t}\rme^{-\rmi t\lambda}$$ the corresponding filter transfer function. The following conditions on the wavelet $\psi$ and scale function $\phi$ are assumed to hold for a positive integer $M$ and a 