Assume we have a Lévy process $L=(L_t)_{t \geq 0}$ and an approximating sequence $(L^n)_{n \in \mathbb{N}} = ((L^n_t)_{t \geq 0})_{n \in \mathbb{N}}$ such that $$ \lim_{n \to \infty} d^{Sk} (L,L^n) = 0 $$ in probability, where $d^{Sk}$ denotes the Skorokhod metric. Alternatively we can say that there is a sequence of change of times functions $(\lambda_n)_{n \in \mathbb{N}} \subset \Lambda$ such that $$ \lim\limits_{n \to \infty} \sup_{s \geq 0} |\lambda_n(s)-s| = 0 $$ and $$ \lim\limits_{n \to \infty} \sup_{s \in [0,N]} |L_s - L^n_{\lambda_n(s)}| = 0 \;\;\; \forall N \in \mathbb{N}, $$ where the latter is limit in probability. Recall that a change of time function is a continuous function $\lambda: [0,\infty) \to [0,\infty)$ which is strictly increasing with $\lambda(0)=0$ and $\lim_{s \to \infty} \lambda(s) = \infty$.
Does it hold that the jumps of $L^n$ converge to the jumps of $L$ too in probability in Skorokhod?
To show this, take the same sequence of change of time functions $\lambda_n$. Denoting the jumps of a process $X$ by $\Delta X_t = X_t - X_{t^-}$ we have for $N \in \mathbb{N}$ and $s \in [0,N]$ that $$ \begin{align*} |\Delta L_s - \Delta L^n_{\lambda_n(s)}| &= |L_s - L_{s^-} -(L^n_{\lambda_n(s)} - L^n_{\lambda_n(s)^-} ) | = |L_s - L^n_{\lambda_n(s)} + (L^n_{\lambda_n(s)^-} -L_{s^-}) | \\ &\leq |L_s - L^n_{\lambda_n(s)}| + |(L^n_{\lambda_n(s)^-} -L_{s^-}) | \end{align*} $$ From the assumption of Skorokhod convergence $L^n$ to $L$ I have that the first term $ \sup |L_s - L^n_{\lambda_n(s)}|$ tends to $0$ as $n \to \infty$. But how to go on with the left limits $|(L^n_{\lambda_n(s)^-} -L_{s^-}) |$?
Some help would be really appreciated or an counterexample, if the statement doesn't even hold.
I thought about applying the continuity of $\lambda$ to get that $\lambda_n(s)^- = \lambda_n(s^-)$. Then I would have $|(L^n_{\lambda_n(s^-)} -L_{s^-}) | \to 0$, since the $s^-$ is the subindex of $L$ and the argument to $\lambda_n$, such that the assumption of Skorokhodconvergence applies again. Is this right?