a probability measure with Laplace exponent $\psi_\mu(u)$, $u>0$. Then the following statements are equivalent.

1.  $\mu\in {\mathrm{T}}$.

2.  $\psi_{\mu_c}(u):=\psi_\mu(u)-\psi_\mu(cu)$ is a CBF for all $c\in(0,1)$.

3.  $-\psi_{\mu_c}(u)= \psi_\mu(cu) - \psi_\mu(u)$ is a CBF for all $c>1$.

4.  $u\cdot \psi_\mu'(u)$ is a CBF.

5.  $\mu={\mathcal{L}}(\int_{(0,\infty)} e^{-t}dX_t)$ for some subordinator $(X_t)_{t\geq 0}$ with ${\mathbb{E}}[\log^+(X_1)]<\infty$ and ${\mathcal{L}}(X_1)\in {\mathrm{BO}}$.

Nested ranges {#sec:nested}
=============

In this section, we will consider what happens with the range $R_\xi^+$ when we modify the characteristics of $\xi$. This result has a counterpart in the case when $\xi$ is a Brownian motion (see [@BLMranges Thm. 5]), although here for some statements we have to restrict on ${\mathrm{L}}\cap R_\xi^+$. That this restriction is truly necessary will subsequently be shown in Proposition \[prop:notnested\].

\[thm:nested\] Let $(\xi_t)_{t\geq 0}$ be a Lévy process with characteristic triplet $(\gamma, \sigma^2, \nu)$ and write $R^+(\gamma, \sigma^2, \nu):=R^+_\xi$.\
Then if $\sigma^2\neq 0$ $$R^+(\gamma, \sigma^2, \nu) = R^+(\gamma/\sigma^2,1, \nu/\sigma^2).$$ Further for $\gamma'\geq \gamma $ it holds $$\label{eq:nested1}
 {\mathrm{L}}\cap R^+(\gamma, \sigma^2, \nu) \subseteq  {\mathrm{L}}\cap R^+(\gamma', \sigma^2, \nu),$$ while assuming that $\nu((0,\infty))=0$ and $\int_{[-1,0)}|x|\nu(dx)<\infty$ we obtain $$\label{eq:nested2}
 R^+(\gamma, \sigma^2, \nu) \subseteq R^+(\gamma', \sigma^2, \lambda\nu)$$ for all $\lambda\in(0,1]$ and $\gamma'$ such that $\gamma'-\gamma \geq -(1-\lambda)\int_{[-1,0)}x\nu(dx)$.

By the Lévy-Itô-decomposition we have $\xi_t=\sigma B_t 