There are also two critical cases that correspond to $\alpha = k - 1/p$ in Theorem \[maintheorem\](i) and $\alpha = k - 1/ \beta$ in Theorem \[maintheorem\](ii). We think that additional logarithmic rates will appear in these cases, but the precise proofs are a subject of future research.

\[rem-const\]

1.  To define the constant $\tilde \sigma$ appearing in Theorem \[sec-order\](i) we set $$\begin{aligned}
    \kappa = {}&\frac{\alpha^{1/(1-\alpha)} }{1-\alpha} 
    \int_0^{\infty} \Phi (y) y^{-1- 1/(1-\alpha )} \,dy,\end{aligned}$$ where $\Phi(y):= {\mathbb{E}}[|\widetilde{X}_1+y|^p - |\widetilde{X}_1|^p]$, $y\in{\mathbb{R}}$, and $\widetilde{X}_t$ is a linear fractional stable motion defined in with $c_0=1$ and $L$ being a standard symmetric $\beta$-stable Lévy process. In addition, set $$\label{def-tau-rho}
     \tau_{\rho}=  \frac{\rho-1}{\Gamma(2-\rho)|\cos(\pi \rho/2)|},\qquad \text{for all }\rho\in (1,2),$$ where $\Gamma$ denotes the gamma function. Then, $$\tilde \sigma = |c_0|^p \sigma^p  \Big(\frac{\tau_\beta}{\tau_{(1-\alpha)\beta}}\Big)^{\frac{1}{(1-\alpha)\beta}}\kappa .$$ The function $\Phi(y)$ can be computed explicitly, see . This representation shows, in particular, that $\Phi (y)>0$ for all $y>0$, and hence the limiting variable $S$ in Theorem \[sec-order\](i) is not degenerate, because $\tilde \sigma >0$.

2.  The constant $\eta^2$ in Theorem \[sec-order\](ii) is given by $$\label{def-eta-23}
    \eta^2 =  |c_0 \sigma |^{2p}\Big(\theta(0) + 2 \sum_{i=1}^\infty \theta(i)\Big)  , \quad
    \theta(i) = a_p^{-2} \int_{{\mathbb{R}}^2} \frac{1}{|s_1s_2|^{1+p}} \psi_i(s_1,s_2)\, ds_1 \,ds_2,$$ $$\begin{aligned}
    \label{theta} \psi_i(s_1,s_2) = {}&  \exp \left(-  \int_{{\mathbb{R}}} 