c_0$ and $\gamma = 1-\alpha$ (indeed, it holds that $1-\alpha\in (1/2,1)$ under conditions of Theorem \[maintheorem\](ii)). In our framework, the function $h$ is given via $h(x)=|x|^p - m_p$, where the quantity $m_p$ has been defined in Theorem \[maintheorem\](ii), which is obviously not bounded. Since $h$ is an even function and $L$ is symmetric, we readily deduce that $k^{\star}=2$.

Now, we summarize the asymptotic theory from [@s04] for the statistic $S_n$ in the case of Appell rank $k^{\star}=2$ ($2<\beta <8/3$):

-   $1/2<\gamma <(\beta + \sqrt{\beta ^2-2\beta})/2\beta$: convergence rate $n^{2-2\gamma}$, Rosenblatt limit.

-   $(\beta + \sqrt{\beta ^2-2\beta})/2\beta<\gamma <2/\beta $: convergence rate $n^{1/\gamma \beta}$, $\gamma \beta$-stable limit.

-   $2/\beta < \gamma < 1$: convergence rate $n^{1/2}$, normal limit.

Although the results of [@s04] are not directly applicable (recall that in our setting $\beta \in (0,2)$ and $h$ is unbounded), Theorem \[sec-order\](i) corresponds to case (ii) of [@s04]. Indeed, we apply a similar proof strategy to show the weak convergence. However, strong modifications due to unboundedness of $h$, triangular nature of summands in , stochastic integrals instead of sums, and the different set of conditions are required.

\[rem1\] Case (ii) of [@s04] is a quite remarkable result, since a $\gamma \beta$-stable distribution appears in 