Z(\lambda)
,\quad t\in\Z \;,$$ hence $Z(\lambda)$ is a Hermitian complex valued process with weakly stationary orthogonal increments on $[-\pi,\pi]$. We further assume that there exist a function $A(u,\lambda)$ in $L^2([0,1]\times[-\pi,\pi])$ and two constants $c>0$ and $D<1/2$ such that $$\label{eq:AssumAtimeSmooth2}
\left|A^0_{t,T}(\lambda)-A(t/T,\lambda)\right|\leq c\; T^{-1} \; |\lambda|^{-D}\,,\quad 1\leq t\leq T,\,-\pi\leq\lambda\leq\pi \;,$$ and $$\label{eq:AssumAtimeSmooth}
\left|A(u;\lambda)-A(v,\lambda)\right|\leq c\; |v-u| \; |\lambda|^{-D}\,,\quad 0\leq u,v\leq1,\,-\pi\leq\lambda\leq\pi\;.$$ These correspond to the definition of locally stationary processes introduced in [@dahlhaus96] but with the term $|\lambda|^{-D}$ added in the upper bound to allow a singularity at the zero frequency. Relations (\[eq:LocStatProcSpectralExt\]), (\[eq:AssumAtimeSmooth2\]) and (\[eq:AssumAtimeSmooth\]) give rise to the following time-varying *generalized* spectral density of $\{X_{t,T}\}$ $$\label{eq:genLocSpecDens}
f(u,\lambda) = |1- \rme^{-\rmi\lambda}|^{-2p}\;\left|A(u;\lambda)\right|^2 \; .$$ The first multiplicative factor in the right-hand side of (\[eq:genLocSpecDens\]) corresponds to the operator $\diffop^p$ in the left-hand side of (\[eq:LocStatProcSpectralExt\]). We now focus on time-varying generalized spectral densities exhibiting a *memory parameter* at zero frequency.

We say that the process $\{X_{t,T},\;t=1, \ldots, T, \ T \geq 1\}$ has *local memory parameter* $d(u)\in(-\infty,p+1/2)$ at time $u\in [0,1]$ if it satisfies (\[eq:LocStatProcSpectralExt\]), (\[eq:AssumAtimeSmooth2\]) and (\[eq:AssumAtimeSmooth\]) and its generalized spectral density $f(u,\lambda)$ defined by (\[eq:genLocSpecDens\]) satisfies the following semi-parametric type condition: $$\label{eq:genLocSpecDensSemiParam}
f(u,\lambda) = |1- \rme^{-\rmi\lambda}|^{-2d(u)}\ f^*(u,\lambda)\ ,$$ with $f^*(u,0)>0$ and $$\label{eq:genLocSpecDensSemiParamCond}
 