with drift $a>0$ and jump measure $\nu$ and set $R^+(a,\nu):=R^+_{\xi}$. Then for $a'>a$ we have $${\mathrm{L}}\cap R^+(a, \nu) \subseteq  {\mathrm{L}}\cap R^+(a', \nu),$$ but $$R^+(a, \nu) \setminus R^+(a', \nu) \neq \emptyset.$$

The first statement has been shown in Theorem \[thm:nested\].\
Let $\mu:=\Phi_{\xi^{(a)}}(\delta_1)$ be the law of $\int_{(0,\infty)} e^{-\xi^{(a)}_t} dt$, then $\mu\in R^+(a, \nu)$ with ${\mathrm{supp}\,}\mu=[0,\frac1a]$ in case of a non-deterministic $\xi$ and ${\mathrm{supp}\,}\mu=\{\frac1a\}$ if $\xi$ is deterministic (cf. [@BLMranges Lemma 2.1]).\
On the other hand by [@BLMranges Lemma 2.1 and Thm. 2.2] all distributions in $R^+(a',\nu)$ have support $[0,\infty)$, $[0, \frac{1}{a'}]$ ($\xi$ non-deterministic) or $\{\frac{1}{a'}\}$ ($\xi$ deterministic). Hence $\mu\not\in R^+(a',\nu)$.

In case of varying jump heights, nested ranges cannot be expected. To illustrate this, we consider the case of Poisson processes with varying jump height in which we can fully describe the range as we shall do in the following proposition, which also improves the previous result [@BLexpfunc Prop. 6.3].

Assume that $\xi_t= c N_t$ for a Poisson process $N=(N_t)_{t\geq 0}$ with intensity $\lambda$ and some $c>0$. Then $$\begin{aligned}
\label{eq-rangePP}
 R_\xi^+& = \{\mu \in {\mathrm{L}}_{e^{-c}} \mbox{ with compound exponentially distributed $e^{-c}$-factor}\} \\
&= \{\mu \in {\mathcal{P}}^+, \mbox{ s.t. } \psi_\mu(u) = \lim_{n\to\infty} \log\left(\frac{\prod_{k=0}^{n-1} (f(e^{-kc} u) +\lambda)}{\lambda^n}\right) \mbox{ for some BF }f 