**Remark 3.**


$$\begin{aligned} W^{(q)}(0) &= \begin{cases} 0, & \text{if X is of unbounded variation,} \\ \frac{1}{c}, & \text{if X is of bounded variation,} \end{cases} \\ W^{(q)\prime}(0+) &:= \lim\_{\mathbf{x}\downarrow 0} W^{(q)\prime}(\mathbf{x}) = \begin{cases} \frac{2}{\sigma^2}, & \text{if } \sigma > 0, \\ \frac{\sigma^2}{\infty}, & \text{if } \sigma = 0 \text{ and } \Pi(0,\infty) = \infty, \\\ \frac{q + \Pi(0,\infty)}{c^2}, & \text{if } \sigma = 0 \text{ and } \Pi(0,\infty) < \infty. \end{cases} \end{aligned}$$

*3. From Lemma 3.3 of Kuznetsov et al. (2013), <sup>W</sup>*Φ(*q*)(*x*) := e<sup>−</sup><sup>Φ</sup>(*q*)*<sup>x</sup> <sup>W</sup>*(*q*)(*x*) *ψ*(Φ(*q*))−1*, as x* ↑ ∞*.*

Due to Remark 3, we make the following assumption throughout the paper.

**Assumption 1.** *We assume that either X has unbounded variation or* Π *is absolutely continuous with respect to the Lebesgue measure. Under this assumption, it holds that W*(*q*) *is* C<sup>1</sup> *in* (0, <sup>∞</sup>)*.*

We give the following properties related to *Z*(*q*) and *W*(*q*) for later use.
