Theorem 36], and hence the above claims follows from the stochastic integral representation $$Z= \int_{[0,1]\times {\mathbb{R}}_0\times {\mathbb{R}}} \big(|c_0 y|^p v\big) \, \Theta(ds,dy,dv).$$ As for Theorem \[maintheorem\](iii), we remark that for values of $\alpha$ close to $k-1/p$ or $k-1/\beta $, the function $g^{(k)}$ explodes at $0$. This leads to unboundedness of the process $F$ defined in Theorem \[maintheorem\](iii). Nevertheless, the limiting random variable in is still finite.

We recall that $L$ is assumed to be a symmetric $\beta$-stable Lévy process in Theorems \[maintheorem\](ii) and \[sec-order\]. This assumption can be relaxed following the discussion of tangent processes in Section \[sec2.1\]. Indeed, only the small scale behaviour of the driving Lévy process $L$ should matter for the statement of the aforementioned results. When the small jumps of $L$ are in the domain of attraction of a symmetric $\beta$-stable Lévy process, e.g.  its Lévy measure satisfies the decomposition $$\nu (dx) = \left(\text{const} \cdot |x|^{-1-\beta} + \varphi(x) \right) dx$$ with $\varphi(x)=o(|x|^{-1-\beta})$ for $x\rightarrow 0$, the statements of Theorems \[maintheorem\](ii) and \[sec-order\] should remain valid under possibly further assumptions on the function $\varphi$. Such processes include for instance tempered or truncated symmetric $\beta$-stable Lévy processes.

\[rem3\] Theorem 5.1 of [@BCI] studies the first order asymptotic of 