\frac{\widehat{V}(x)}{\widehat{V} \big( h^{-1}(1/t) \big)} - \frac{\widehat{V}(x)V(R)}{t} {\leqslant}{\mathbb{P}}^x \big( {\tau_{(0,R)}}< {\tau_{(0,\infty)}}\big).$$

Now we specify $t>0$. Set $t=a/h(R)$, where $a {\geqslant}1$. By [@GS2019 Lemma 2.3], there is $c_2 {\geqslant}1$ such that $$h^{-1}(1/t)=h^{-1} \big( h(R)/a \big) {\leqslant}c_2 a^{1/\alpha}R.$$ Taking into account subadditivity and monotonicity of $\widehat{V}$, we infer that $$\widehat{V} \big( h^{-1}(1/t) \big) {\leqslant}\widehat{V} \big( c_2 a^{1/\alpha}R \big) {\leqslant}2c_2 a^{1/\alpha} \widehat{V}(R),$$ if only $a {\geqslant}(2c_2)^{-\alpha}$. Furthermore, by [@TG19 Corollary 5], there is $c_3 {\geqslant}1$ such that $$h(R) {\leqslant}\frac{c_3}{V(R)\widehat{V}(R)}.$$ It follows that $$\begin{aligned}
    {\mathbb{P}}^x \big( {\tau_{(0,R)}}< {\tau_{(0,\infty)}}\big) &{\geqslant}2^{-1}c_1 c_2^{-1} a^{-1/\alpha} \frac{\widehat{V}(x)}{\widehat{V}(R)} - \frac{c_3}{a} \frac{\widehat{V}(x)}{\widehat{V}(R)} = \frac{\widehat{V}(x)}{\widehat{V}(R)} \frac1a \Big( 2^{-1}c_1 c_2^{-1}a^{(\alpha-1)/\alpha} - c_3 \Big) \\ &{\geqslant}\frac{1}{a} \frac{\widehat{V}(x)}{\widehat{V}(R)},
    \end{aligned}$$ if $a$ is big enough, and the claim follows.

Regular variation {#sec:regvar}
=================

In this section we aim to prove that the regular variation of the real part of the characteristic exponent implies regular variation of its imaginary part, if we impose some condition on the structure of the Lévy measure. First, we recall some basic definitions and properties. The general reference here is the book [@RegularVariation89].

We say that a function $f\colon (0, \infty) \mapsto (0, \infty)$ is regularly varying at $0$ with parameter $\rho$, if for all $\lambda>0$, $$\begin{aligned}
\lim_{x \to 0^+} \frac{f(\lambda x)}{f(x)} = 