Theorem \[maintheorem\](i) seem to be sharp. Indeed, since $|h_k(x)|\leq Kx^{\alpha -k}$ for large $x$, we obtain from that $$\sup_{m\geq 1} V_m <\infty$$ when $\alpha \in (0,k-1/p)$. On the other hand $\sum_{m:T_m\in [0,1]} |\Delta L_{T_m}|^p<\infty$ for $p>\beta$, which follows from the definition of the Blumenthal–Getoor index at . Notice that under assumption $\alpha \in (0, k-1/2)$ the case $p=2$, which corresponds to quadratic variation, always falls under Theorem \[maintheorem\](i). We remark that the distribution of the limiting variable in does not depend on the chosen sequence $(T_m)_{m\geq 1}$ of stopping times which exhausts the jump times of $L$. Furthermore, the limiting random variable $Z$ in is infinitely divisible with Lévy measure $(\nu\otimes \eta)\circ \big((y,v)\mapsto  |c_0 y|^p v\big)^{-1}$, where $\eta$ denotes the law of $V_1$. In fact, $Z$ has characteristic function given by $${\mathbb{E}}[\exp(i \theta Z)]=\exp\Big( \int_{{\mathbb{R}}_0\times {\mathbb{R}}} (e^{i \theta |c_0 y|^p v}-1)  \,\nu(dy)\,\eta(dv)   \Big).$$ To show this, let $\Lambda$ be the Poisson random measure given by $\Lambda=\sum_{m=1}^\infty \delta_{(T_m,\Delta L_{T_m})}$ on $[0,1]\times {\mathbb{R}}_0$ which has intensity measure $\lambda\otimes \nu$. Here ${\mathbb{R}}_0:={\mathbb{R}}\setminus \{0\}$ and $\lambda$ denotes the Lebesgue measure on $[0,1]$. Set $\Theta=\sum_{m=1}^\infty \delta_{(T_m, \Delta L_{T_m},V_m)}$. Then $\Theta$ is a Poisson random measure with intensity measure $\lambda\otimes \nu \otimes \eta$, due to [@Serfozo 