that $ 1 - \sum_{k=1}^q \phi_k(u) z^k$ does not vanish for all $u\in[0,1]$ and $z\in\C$ such that $|z|\leq1$. Using this latter condition, the local transfer function $A(u;\cdot)$ defines a causal *autoregressive fractionally integrated moving average* (ARFIMA$(q,d(u)-p,r)$) process and the local generalized spectral density (\[eq:genLocSpecDens\]) satisfies the conditions (\[eq:genLocSpecDensSemiParam\]) and (\[eq:genLocSpecDensSemiParamCond\]) with $\beta=2$. Using Lemma \[lem:lipshitz\_with\_log\] in \[sec:technical-lemmas\], the Lipschtiz assumptions on $d$, $\sigma$, $\theta$ and $\phi$ yield the condition (\[eq:AssumAtimeSmooth\]) with $D>\sup_u d(u)-p$.

In order to verify Condition (\[eq:AssumAtimeSmooth2\]) trivially, the simplest definition of $\{\diffop^pX_{t,T}\}$ in all the previous examples is to take $A^0_{t,T}(\lambda)=A(t/T,\lambda)$, that is to set the time-varying linear representation $$\label{eq:LocStatProcSpectral}
\diffop^p X_{t,T} = 
\int_{-\pi}^\pi A(t/T;\lambda)\ \rme^{\rmi\lambda t}\;\rmd Z(\lambda)\;,$$ as will be done for our simulated tvARFIMA in Section 5. However, one might also want to use a different transfer function $A^0_{t,T}$ in (\[eq:LocStatProcSpectralExt\]), provided that Condition (\[eq:AssumAtimeSmooth2\]) holds. Such approximated time varying linear representation is motivated by the tvAR($p$) process, which satisfies the recursion $$X_{t,T}-\sum_{k=1}^p\phi_k(t/T)X_{t-k,T}=\sigma(t/T)\;\varepsilon_t, \quad 1\leq t\leq T \;,$$ along with appropriate initial conditions. It has been shown in [@dahlhaus96] that such non-stationary process does not satisfy a representation of the form (\[eq:LocStatProcSpectral\]) (with $p=0$) but it does satisfy (\[eq:LocStatProcSpectralExt\]) and (\[eq:AssumAtimeSmooth2\]) (with $p=D=0$).

Estimation method 