372, l. 11), and this result agrees with our Theorem \[maintheorem\](i). However, in the last three lines of their proof (p. 372, l. 12–15) it is argued how ${\mathbb{E}}[  2^{n\alpha p} V(p;2)_{2^n}] \to C$ implies that $2^{n\alpha p} V(p;2)_{2^n}\to C$ a.s. The argument relies on the statement from [@BCI]: “$S_n(y)= {\mathbb{E}}[ S_n(y) ] (1+o_{(a.s.)}(1))$”, where the stochastic process $S_n(y)$ is the empirical characteristic function of normalised increments defined on [@BCI p. 367]. This statement is only shown for each fixed $y$ (see [@BCI Lemma 5.2]), but to use it to carry out the proof, it is crucial to know the dependence of the $o_{(a.s.)}(1)$-term in $y$. In fact, it is not even enough to have boundedness in $y$ of the $o_{(a.s.)}(1)$-term.

Let us further explain the various conditions of Theorems \[maintheorem\](ii) and \[sec-order\]. The condition $p<\beta$ obviously ensures the existence of moments $m_p$, while assumption $p<\beta/2$ ensures the existence of variance of the statistic $V(p;k)_n$. The validity range of the central limit theorem ($\alpha \in (0,k-2/\beta)$) in is smaller than the validity range of the law of large numbers in Theorem \[maintheorem\](ii) ($\alpha \in (0,k-1/\beta)$). It is not clear which limit distribution appears in case of $\alpha \in (k-2/\beta, k-1/\beta)$. 