is regularly varying at $0+$ with index $-\rho$, for some $\rho>0$ that satisfies $$\begin{aligned}
\label{Equation assumption rho}
            \max(2\omega_--\omega_+,-\alpha)<\rho<\omega_-.
        \end{aligned}$$ This is equivalent to regular variation with index $\omega_--\rho$ of the tail $x\mapsto\Pi^-({\left(-\infty\,,\log(x)\right)})$ as $x\to 0+$. Indeed, suppose holds and recall that by , we have that $$\begin{aligned}
            \Pi^-({\left(-\infty\,,\log(x)\right)})
            &=\int_{{\left(-\infty\,,\log(x)\right)}}e^{\omega_- y}\Lambda(\mathrm{d}y)+\int_{{\left(\log(1-x)\,,0\right)}}(1-e^{y})^{\omega_-}\Lambda(\mathrm{d}y).
    \end{aligned}$$ We see that $$\begin{aligned}
        \int_{{\left(-\infty\,,\log(x)\right)}}e^{\omega_- y}\Lambda(\mathrm{d}y)
        \leq x^{\omega_-}\overline{\Lambda}(\log(x))
        =o(x^{\omega_-}). 
    \end{aligned}$$ To estimate the second integral, we use and Theorem 1.6.5 in [@BGT87], which then gives us that $$\begin{aligned}
\label{Equation tail Pi^-}
        \Pi^-({\left(-\infty\,,\log(x)\right)})
        &\underset{x\to 0+}{\sim}\int_{{\left(\log(1-x)\,,0\right)}}(1-e^{y})^{\omega_-}\Lambda(\mathrm{d}y)
        \underset{x\to 0+}{\sim}\overline{\Lambda}\left(x\right)x^{\omega_-}\frac{\rho}{\omega_--\rho}.
    \end{aligned}$$ The converse also follows from the same theorem, that is if $x\mapsto\Pi^-({\left(-\infty\,,\log(x)\right)})$ has regular variation at $0+$ with index $\omega_--\rho$ then it is also the case for $\overline{\Lambda}$ with index $-\rho$. Indeed, coming back to the first part of and using $1-e^{-y}\sim y$ and $\log(1-y)\sim y$ as $y\to 0+$, we see that $$\begin{aligned}
        \Pi^-({\left(-\infty\,,\log(x)\right)})
        &\underset{x\to 0+}{\sim}\int_{{\left(-x\,,0\right)}}|y|^{\omega_-}\Lambda(\mathrm{d}y).
    \end{aligned}$$ This is enough to conclude using [@BGT87] Theorem 1.6.5 (see the discussion following the proof of the theorem).

Similarly, one can show that regular variation with index $-\rho$ of $\overline{\Lambda}$ at $0+$ is equivalent to regular variation with index $\omega_+-\rho$ of $x\mapsto\Pi^+({\left(-\infty\,,\log(x)\right)})$ as $x\to 0+$.

Recall the definition of $I$ from .

\[Theorem area GF\] 