|s_1 h_k(x) -s_2h_k(x+i)|^{\beta } \,dx \right)  
      \\[1.5 ex]
    {}&- \exp \left(-  \int_{{\mathbb{R}}} |s_1 h_k(x)|^{\beta} + |s_2h_k(x+i)|^{\beta} \,dx \right), \nonumber\end{aligned}$$ where the function $h_k$ is defined at and $a_p:= \int_{{\mathbb{R}}} (1-\exp(iu)) |u|^{-1-p} \,du$.

\[rem4\] Let us explain the somewhat complex form of the variance $\eta^2$ in . A major problem of proving Theorems \[sec-order\](ii) is that the covariance structure of stationary summands of $V(p;k)_n$ can not be computed directly. However, the identity $$\begin{aligned}
 \label{xp}
|x|^p = a_p^{-1} \int_{{\mathbb{R}}} (1-\exp(iux)) |u|^{-1-p} du \qquad \text{for } p\in (0,1),\end{aligned}$$ which can be shown by substitution $y=ux$ ($a_p$ is defined in Remark \[rem-const\](ii)), turns out to be a useful instrument. Indeed, for any deterministic function $\varphi :{\mathbb{R}}\rightarrow {\mathbb{R}}$ satisfying $\varphi\in L^{\beta}({\mathbb{R}})$, it holds that $$\begin{aligned}
 \label{charfun}
{\mathbb{E}}\left[ \exp \left(iu \int_{{\mathbb{R}}} \varphi(s) \,dL_s \right)\right] = \exp \left(-\sigma^{\beta}|u|^{\beta} \int_{{\mathbb{R}}} |\varphi(s)|^{\beta} \,ds \right),\end{aligned}$$ where $\sigma>0$ is the scale parameter of $L$. These two identities are used to compute the variance of the statistic $V(p;k)_n$ and they are both reflected in the formula for the quantity $\theta(i)$ in .

\[rem6\] A key to the proof of Theorem \[sec-order\](ii) is a quite technical Lemma \[helplem2\], which gives an upper bound for the covariances of the summands of $V(p;k)_n$. We do 