In Latz (2021), the space of continuous paths \(C^0([0, \infty ); X)\) needs to be equipped with the metric given by

$$\begin{aligned}{} & {} \rho (x,y):= \sum _{n=1}^\infty 2^{-n}\min \left\{ 1,\sup _{s \in [0,n]}\Vert x(s)-y(s)\Vert \right\} \\{} & {} \qquad (x,y \in C^0([0, \infty ); X)) \end{aligned}$$

rather than the supremum norm \(\Vert x\Vert _\infty = \sup _{t \in [0,\infty )}\Vert x(t)\Vert \), see Kushner (1984). This has the following implications:

(1) The convergence results in Theorem 1 and Lemma 2 are now weaker, as the metric \(\rho \) is weaker than the metric induced by \(\Vert \cdot \Vert _\infty \).

(2) Proposition 4(i) does not hold uniformly in time. The function F in the proof of this proposition can be chosen as \(F((\xi (t), \tau (t), {\varvec{j}}(t))_{t \ge 0},t):= \min \{1, \Vert \xi (t)\Vert \}2^{-(t+1)}\) – it now depends on t. This F is bounded and Lipschitz continuous in \((\xi (t))_{t \ge 0}\) with respect to \(\rho \), continuous in \(t>0\) and constant with respect to the other inputs. To show Lipschitz continuity, we observe

$$\begin{aligned}&|F((\xi (t), \tau (t), {\varvec{j}}(t))_{t \ge 0},t)- F((\xi '(t), \tau '(t), {\varvec{j}}'(t))_{t \ge 0},t)| \\&\quad = | \min \{1, \Vert \xi (t)\Vert \} - \min \{1, \Vert \xi '(t)\Vert \}|2^{-(t+1)} \\&\quad \le \min \{1, \Vert \xi (t)-\xi '(t)\Vert \}2^{-(t+1)}\\&\le \rho ((\xi (t))_{t \ge 0}, (\xi '(t))_{t \ge 0}) \end{aligned}$$

for \(t > 0\), \((\xi (t))_{t \ge 0}, (\xi '(t))_{t \ge 0}, (\tau (t))_{t \ge 0}, (\tau '(t))_{t \ge 0} \in C^0([0, \infty ); X)\) and \(({\varvec{j}}(t))_{t \ge 0}, ({\varvec{j}}'(t))_{t \ge 0}\) being Markov jump processes on I. Thus, the weak convergence shown in Lemma 2 now implies that

$$\begin{aligned}{} & {} \mathbb {E}[F((\xi _\varepsilon (t) - \xi (t), \tau _\varepsilon (t) -\tau (t), {\varvec{j}}_\varepsilon (t)\\{} & {} - {\varvec{j}}(t))_{t \ge 0},t)] \rightarrow 0 \qquad (\varepsilon \downarrow 0; t > 0) \end{aligned}$$

and the inequality in Proposition 4(i) now reads

$$\begin{aligned} \textrm{W}_1(\textrm{D}^{\varepsilon }_{t|0}(\cdot |\xi _0, j_0),\textrm{D}_{t|0}(\cdot |\xi _0, j_0)) \le \alpha '(\varepsilon ,t), \end{aligned}$$

with \(\alpha (0,t) = 0\) and \(\varepsilon \mapsto \alpha (\varepsilon , t)\) is continuous at 0 for any \(t>0\).

We also note that this \(\alpha '\) can depend on the initial values \(\xi _0\) and \(j_0\) and that the definition of \(t_0\) in this proposition is not necessary.

(3) Theorem 4 is still correct as stated with a refined argument in the proof. In the last inequality of the proof of Theorem 4, we can choose \(\varepsilon _t\) to depend on t. The continuity of \(\alpha '\) and \(\alpha ''\) implies that for any \(\delta > 0\), we can find \(\varepsilon _t > 0\) small enough such that \(\alpha '(\varepsilon _t,t) + \alpha ''(\varepsilon _t) \le \delta \) for all \(t > 0\), which is sufficient to prove convergence as stated in Theorem 4.