the limit although the summands of $S_n$ are bounded (in particular, all moments of $S_n$ exist). In particular, the rate of convergence $n^{1/\gamma \beta}$ does not correspond to the variance of $S_n$.

\[rem2\] The symmetry condition on the Lévy process $L$ is assumed for sake of assumption simplification. Most asymptotic results of this paper would not change if we dropped this condition. However, the Appell rank of the function $h(x)=|x|^p - m_p$ might be $1$ when $L$ is not symmetric and this does change the result of Theorem \[sec-order\](i). More specifically, the limiting distribution turns out to be $\beta$-stable (see e.g.  [@ks01] for the discrete case). We dispense with the exact exposition of this case.

Further remarks and possible extensions {#sec2.4}
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We start by commenting on the set of conditions introduced in assumption (A). First of all, it follows by [@RajRos Theorem 7] that the process $X$, defined in , is well-defined if and only if for all $t\geq 0$, $$\label{sdkfhsd;kf}
\int_{-t}^\infty \int_{\mathbb{R}}\Big(\big| f_t(s)x\big|^2\wedge 1\Big)\,\nu(dx)\,ds<\infty,$$ where $f_t(s)=g(t+s)-g_0(s)$. By adding and subtracting $g$ to $f_t$ it follows by assumption (A) and the mean value theorem that $f_t\in L^\theta({\mathbb{R}}_+)$ and $f_t$ is bounded. For all $\epsilon>0$, assumption (A) implies that $$\label{eq:74}
