the corresponding transfer function is not explicitly given and thus (\[eq:AssumAtimeSmooth\]) is difficult to check. To circumvent this problem, one may construct an example by approximating a causal continuous time representation of the FBM, see *e.g.* [@samo:2006 Chapter 6]. Let us fix $H$ in $(1/2,1)$. Replacing the integral by a discrete sum in this representation, one obtains the following process $$\tilde{B}_H(t)=\sum_{s\in\Z}\{(t-s)_+^{H-1/2}-(-s)_+^{H-1/2}\}\,\varepsilon_s,\quad t\in\Z\;,$$ where $\{\varepsilon_s\}_{s\in\Z}$ is a standard Gaussian white noise. Then $$\diffop \tilde{B}_H(t) =
\sum_{k\geq0}\{k^{H-1/2}-(k-1)_+^{H-1/2}\}\,\varepsilon_{t-k},\quad t\in\Z\;,$$ is a causal representation of a stationary process. Using an integral approximation of the discrete Fourier transform of the sequence $(k_+^{H-1/2}-(k-1)_+^{H-1/2})_{k\in\Z}$, one can show that, as $\lambda\to0$, the corresponding transfer function $A_H(\lambda)$ satisfies $|A_H(\lambda)|=C_H\,|\lambda|^{1/2-H}+O(1)$ for some positive constant $C_H$. Moreover, for any $\epsilon>0$, there is a constant $C$ such that, for all $1/2<H'\leq H<1$ and $\lambda\in(-\pi,\pi)$, $$|A_H(\lambda)-A_{H'}(\lambda)|\leq C\,|H-H'|\,|\lambda|^{H-1/2-\epsilon}\;.$$ Let now $H$ be a Lipschitz mapping of $[0,1]$ into a subset of $(1/2,1)$ and define the approximated causal tvFBM($H$) process by setting $A(u,\lambda)=A_{H(u)}(\lambda)$. Then Condition (\[eq:AssumAtimeSmooth\]) holds again for any $D>\sup_u H(u)-1/2$.

The *time varying autoregressive fractionally integrated moving average* (tvARFIMA$(q,d,r)$) process is defined by $$\label{eq:arfimaLocSpecDens}
A(u;\lambda)=(1-\rme^{-\rmi\lambda})^{-d(u)+p}\;
\frac{\sigma(u)}{\sqrt{2\pi}}\;
\frac{1+\sum_{k=1}^r\theta_k(u) \rme^{-\rmi\lambda}}
{1 - \sum_{k=1}^q \phi_k(u) \rme^{-\rmi\lambda}}\;,$$ where $d:[0,1]\to(-\infty,p+1/2)$, $\sigma:[0,1]\to\R_+$, $\phi=[\phi_1\,\,\dots\,\,\phi_q]^T:[0,1]\to\R^q$ and $\theta=[\theta_1\,\,\dots\,\,\theta_r]^T:[0,1]\to\R^r$ are Lipschitz functions such 