$X$ is well-defined, cf. Section \[sec2.4\]. When $L$ is a $\beta$-stable Lévy process, we always choose $\theta = \beta$ in assumption (A). Before we introduce the main results, we need some more notation. Let $h_k\!:{\mathbb{R}}\to{\mathbb{R}}$ be given by $$\begin{aligned}
\label{def-h-13}
h_k(x)&=  \sum_{j=0}^k (-1)^j \binom{k}{j} (x-j)_{+}^{\alpha},\qquad x\in {\mathbb{R}},\end{aligned}$$ where $y_+=\max\{y,0\}$ for all $y\in {\mathbb{R}}$. Let ${\mathbb{F}}=({\mathcal F}_t)_{t\geq 0}$ be the filtration generated by $(L_t)_{t\geq 0}$, $(T_m)_{m\geq 1}$ be a sequence of ${\mathbb{F}}$-stopping times that exhaust the jumps of $(L_t)_{t\geq 0}$, that is, $ \{T_m(\omega):m\geq 1\}\cap {\mathbb{R}}_+ = \{t\geq 0: \Delta L_t(\omega)\neq 0\}$ and $T_m(\omega)\neq T_n(\omega)$ for all $m\neq n$ with $T_m(\omega)<\infty$. Let $(U_m)_{m\geq 1}$ be independent and uniform $[0,1]$-distributed random variables, defined on an extension $(\Omega', \mathcal F', \mathbb P')$ of the original probability space, which are independent of ${\mathcal F}$.

The following two theorems summarize the first and second order limit theory for the power variation $V(p;k)_n$. We would like to emphasize part (i) of Theorem \[maintheorem\] and part (i) of Theorem \[sec-order\], which are truly remarkable probabilistic results. We refer to [@Aldous; @ren] and to Section \[proofs-w3lkhj\] for the definition of $\mathcal F$-stable convergence which will be denoted ${\stackrel{\mathcal{L}-s}{\longrightarrow}}$.

\[maintheorem\] Suppose (A) is satisfied and assume that the Blumenthal–Getoor 