adapted from [@moulines:roueff:taqqu:2007:jtsa] to the time varying setting.

The *Fractional Brownian motion* (FBM) process $\{ B_H(k) \}_{k \in \Z}$ with Hurst index $H\in(0,1)$ is a discrete-time version of $\{ B_H(t), t \in
  \R \}$, a Gaussian process with mean zero and covariance $$\E[ B_H(t) B_H(s)] = \frac{1}{2} \left\{ |t|^{2H} + |s|^{2H} - |t-s|^{2H} \right\} \; .$$ The spectral density of $\{ \diffop B_H(k) \}_{k \in \Z}$ is then given by $\lambda\mapsto |1-\rme^{-\rmi\lambda}|^{-2H+1}f_{\mathrm{FBM}}(\lambda;H)$, where $$\label{eq:smooth:part:FBM}
f_{\mathrm{FBM}}(\lambda;H)=\left|\frac{2 \sin(\lambda/2)}{\lambda}\right|^{2H+1}+
\left|2\sin(\lambda/2)\right|^{2H+1}\sum_{k\neq0}\left|\lambda+2k\pi\right|^{-2H-1}\;.$$ The *time varying Fractional Brownian motion* (tvFBM) has generalized spectral density (\[eq:genLocSpecDensSemiParam\]) with $p=1$, $d(u)=H(u)+1/2\in(1/2,3/2)$ and $f^\ast(u,\lambda)=f_{\mathrm{FBM}}(\lambda;H(u))$, where $H$ is a Lipschitz mapping of $[0,1]$ into a subset of $(0,1)$. Then (\[eq:genLocSpecDensSemiParamCond\]) holds with $\beta=(2H(u)+1)\wedge2$. The corresponding non-negative local transfer function is $$\label{eq:tvFBMkernel}
A(u,\lambda)=|1- \rme^{-\rmi\lambda}|^{1/2-H(u)}\sqrt{f_{\mathrm{FBM}}(\lambda;H(u))}\;.$$ In this case, by Lemma \[lem:lipshitz\_with\_log\] in \[sec:technical-lemmas\], (\[eq:AssumAtimeSmooth\]) holds for any $D>\sup_u H(u)-1/2$.

The *time varying fractional Gaussian noise* (tvFGN) is defined similarly as the tvFBM by $f^\ast(u,\lambda)=f_{\mathrm{FBM}}(\lambda;H(u))$ but with $p=0$ and $d(u)=H(u)-1/2\in(-1/2,1/2)$.

The drawback of the tvFBM (and also of tvFGN) defined above is the non-causality of the transfer function $A(u,\cdot)$ defined in (\[eq:tvFBMkernel\]). Since $\{ \diffop B_H(k)
  \}_{k \in \Z}$ is purely non-deterministic, it admits a causal representation. On the other hand, to our knowledge, 