variation of ${\operatorname{Re}}\psi$ at 0 and regular variation of the tail of $\nu$ at infinity can be easily obtained from [@MR0231423]. Our proof, however, works in both cases.

If the Lévy measure is of the special form $$\begin{aligned}
\nu(dx) = C_d \mathds{1}_{\{x<0\}}\nu_0(dx) + C_u \mathds{1}_{\{x>0\}}\nu_0(dx), \label{gest-miar}\end{aligned}$$ where $\nu_0(dx)$ is a symmetric Lévy measure, the theorem above provides the behaviour of the one-sided tail of $\nu$ as well. For instance, if ${\operatorname{Re}}\psi$ is regularly varying at $0$ with the exponent $\alpha \in (0,2)$, then $$\begin{aligned}
\nu_0([t,\infty))  \cong \frac{1}{C_u + C_d}  \frac{\Gamma(1 + \alpha) }{\mathrm{B}\Big(1 - \frac{\alpha}{2},1 + \frac{\alpha}{2}\Big)} {\operatorname{Re}}\psi(1/t), \quad t \to \infty.\end{aligned}$$ This is the case for stable processes, where even the equality holds true.

\[finite\_moment\] If ${\operatorname{Re}}\psi$ varies regularly at $0$ with an exponent $\alpha>1$, then $$\begin{aligned}
\int_{(-1,1)^c} |x| \,\nu(dx) < \infty.\end{aligned}$$

If ${\operatorname{Re}}\psi$ varies regularly at $0$ with a positive exponent ${\operatorname{Re}}\psi\cong \psi^*$ near $0$. Hence $h(r)\approx {\operatorname{Re}}\psi(1/r)$ for large $r>R_0$. By the Potter Lemma we get $$\begin{aligned}
\int_{|x|>1}|x|\,\nu(dx)&=\int^\infty_0\nu\big(\{s\colon|s|>1\vee u\}\big)\,du \\&{\leqslant}\int^\infty_0 h(1\vee u)\,du{\leqslant}h(1)R_0+c\int^\infty_{R_0}{\operatorname{Re}}\psi(1/u)\,du<\infty.\end{aligned}$$ The claim follows immediately.

\[lem:measure\_function\] Assume that $f(s)$, $s {\geqslant}0$, is a function which is non-negative, regularly varying at $\infty$ with parameter $-\alpha,$ where $\alpha \in (1,2)$, and such that $\int_0^\infty (1 \wedge s^2)f(s)\,ds < 