index satisfies $\beta<2$. We obtain the following three cases:

-   \[mt-case-1\] Suppose that (A-log) holds if $\theta=1$. If $\alpha <k-1/p$ and $p>\beta$ we obtain the ${\mathcal F}$-stable convergence $$\label{part1}
    n^{\alpha p}V(p;k)_n {\stackrel{\mathcal{L}-s}{\longrightarrow}}|c_0|^p\!\!\!\!\! \!\sum_{m:\, T_m\in [0,1]} |\Delta L_{T_m}|^p V_m\quad\text{where}\quad 
    V_m=\sum_{l=0}^{\infty} |h_k(l+U_m)|^p.$$

-   \[mt-case-2\] Suppose that $L$ is a symmetric $\beta$-stable Lévy process with scale parameter $\sigma>0$. If $\alpha <k-1/\beta$ and $p<\beta$ then it holds $$\begin{aligned}
     \label{part2}
    n^{-1+p(\alpha + 1/\beta)}V(p;k)_n {\stackrel{{\mathbb{P}}}{\longrightarrow}}m_p\end{aligned}$$ where $m_p=|c_0|^p \sigma^p (\int_{\mathbb{R}}|h_k(x)|^\beta\,dx)^{p/\beta}{\mathbb{E}}[|Z|^p]$ and $Z$ is a symmetric $\beta$-stable random variable with scale parameter $1$.

-   \[mt-case-3\] Suppose that $p\geq 1$. If $p= \theta$ suppose in addition that (A-log) holds. For all $\alpha>k-1/( \beta\vee p)$ we deduce $$\label{part3}
      n^{-1+pk}V(p;k)_n {\stackrel{{\mathbb{P}}}{\longrightarrow}}\int_0^1 |F_u|^p\, du$$ where $(F_u)_{u\in {\mathbb{R}}}$ is a measurable process satisfying $$F_u=  \int_{-\infty}^u g^{(k)}(u-s) \,dL_s\quad \text{a.s.\ for all }u\in {\mathbb{R}}\quad \text{and}\quad \int_0^1 |F_u|^{p}\,du<\infty\quad \text{a.s.}$$

We remark that, except the critical cases where $p= \beta$, $\alpha= k-1/p$ and $\alpha= k-1/\beta$, Theorem \[maintheorem\] covers all possible choices of $\alpha>0,\beta\in [0,2)$ and $p\geq 1$. We also note that the limiting random variable in is infinitely divisible, see Section \[sec2.4\] for more details. In addition, we note that there is no convergence in probability in due to the fact that the random variables $V_m$, $m\geq 