\big\{g(t-s)-g_0(-s)\big\}\, dL_s,\end{aligned}$$ where $L=(L_t)_{t\in {\mathbb{R}}}$ is a symmetric Lévy process on ${\mathbb{R}}$ with $L_0=0$, that is, for all $u\in {\mathbb{R}}$, $(L_{t+u}-L_u)_{t\geq 0}$ is a Lévy process indexed by ${\mathbb{R}}_+$ which distribution is invariant under multiplication with $-1$. Furthermore, $g$ and $g_0$ are deterministic functions from ${\mathbb{R}}$ into ${\mathbb{R}}$ vanishing on $(-\infty,0)$. In the further discussion we will need the notion of *Blumenthal–Getoor index* of $L$, which is defined via $$\begin{aligned}
 \label{def-B-G}
\beta:=\inf\Big\{r\geq 0: \int_{-1}^1 |x|^r\,\nu(dx)<\infty\Big\}\in [0,2],\end{aligned}$$ where $\nu$ denotes the Lévy measure of $L$. When $g_0=0$, process $X$ is a moving average, and in this case $X$ is a stationary process. If $g(s)=g_0(s)=s^\alpha_+$, $X$ is a so called . In particular, when $L$ is a $\beta$-stable Lévy motion with $\beta \in (0,2)$, $X$ is called a linear fractional stable motion and it is self-similar with index $H=\alpha+1/\beta$; see e.g. [@SamTaq] (since in this case the stability index and the Blumenthal–Getoor index coincide, they are both denoted by $\beta$).

Probabilistic analysis of stationary increments Lévy driven moving averages such as semimartingale property, fine scale structure and integration concepts, have been investigated in several papers. We refer to the work of [@BasRosSM; @BCI; @bls12; @bm08; @FK] among many others. However, only few 