\Big) {\stackrel{d}{\longrightarrow}}\mathcal N(0, \eta^2),\end{aligned}$$ where the quantity $\eta^2$ is defined in Remark \[rem-const\](ii).

This paper is structured as follows. Section \[sec2\] presents some remarks about the nature and applicability of the main results, and it also includes a discussion about related problems. Section \[secPrel\] introduces some preliminaries. We state the proof of Theorem \[maintheorem\] in Section \[proofs-w3lkhj\], while the proof of Theorem \[sec-order\] is demonstrated in Section \[sec5\]. Some technical lemmas are deferred to the Appendix.

Related results, remarks and extensions {#sec2}
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In this section we will give a review of related asymptotic results in the literature, present some intuition behind our limit theory and discuss possible extensions.

Fine properties of stationary increments Lévy driven moving averages {#sec2.1}
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In this subsection we will discuss the probabilistic properties of the process $X$ defined at , such as semimartingale property and small scale behaviour, and their consequences for the limit theory. For simplicity suppose that $L$ is a symmetric $\beta$-stable Lévy process.

Suppose, in addition, that $g'$ satisfies the lower bound $K t^{\alpha-1}\leq |g'(t)|$ for all $t\in (0,1)$ and a constant $K>0$. By [@BasRosSM Example 4.9] it follows that $X$ is a semimartingale if and only if $\alpha>1-1/\beta$, which is 