exactly condition (iii) in Theorem \[maintheorem\] when $k=1$ and $p<\beta$.

To better understand the limit theory stated in Theorem \[maintheorem\](ii) and Theorem \[sec-order\], which both refer to the ergodic case, we need to study the small scale behaviour of the stationary increments Lévy driven moving averages $X$. We intuitively deduce the following approximation for the increments of $X$ for a small $\Delta>0$: $$\begin{aligned}
X_{t+\Delta} - X_t &= \int_{{\mathbb{R}}} \{g(t+\Delta -s) - g(t -s)\}\, dL_s \\
& \approx \int_{t+\Delta -\epsilon }^{t+\Delta} \{g(t+\Delta -s) - g(t -s)\} \,dL_s \\
& \approx c_0 \int_{t+\Delta -\epsilon }^{t+\Delta} \{(t+\Delta -s)_{+}^\alpha  - (t -s)_{+}^\alpha \} \,dL_s \\
& \approx c_0 \int_{{\mathbb{R}}} \{(t+\Delta -s)_{+}^\alpha  - (t -s)_{+}^\alpha \} \,dL_s = \widetilde{X}_{t+\Delta} - \widetilde{X}_t,\end{aligned}$$ where $$\begin{aligned}
 \label{flm1}
\widetilde{X}_t := c_0 \int_{{\mathbb{R}}} \{(t-s)_{+}^\alpha  - ( -s)_{+}^\alpha \}\, dL_s,\end{aligned}$$ and $\epsilon >0$ is an arbitrary small real number with $\epsilon \gg \Delta$. In the classical terminology $\widetilde{X}$ is called the *tangent process of $X$*. The formal proof of this first order approximation, which will be demonstrated in Section \[proofs-w3lkhj\], relies on assumption (A) and the fact that, under conditions of Theorem \[maintheorem\](ii), the weight $g(t+\Delta -s) - g(t -s)$ attains asymptotically highest values when $s\approx t$, since $g'$ explodes at 