Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$ and $(X_t)_{t\ge0}$ be an $E$-valued continuous process on $(\Omega,\mathcal A,\operatorname P)$ with $$\left(X^n_{t_0},\ldots,X^n_{t_k}\right)\xrightarrow{n\to\infty}\left(X_{t_0},\ldots,X_{t_k}\right)\tag1$$ in distribution for all $k\in\mathbb N_0$ and $0=t_0<\cdots<t_k$.

Are we able to conclude $$X^n\xrightarrow{n\to\infty}X\tag2$$ in distribution (with respect to the Skorohod topology)?

I know that if $S$ is a Polish space, then a sequence $(\mu_n)_{n\in\mathbb N}$ of probability measures on $(S,\mathcal B(S))$ converges weakly to a probability measure $\mu$ on $(S,\mathcal B(S))$ if and only if $(\mu_n)_{n\in\mathbb N}$ is tight and there is a separating family $\mathcal C$ of bounded continuous functions $S\to\mathbb R$ such that $$\int f\:{\rm d}\mu=\lim_{n\to\infty}\int f\:{\rm d}\mu_n\;\;\;\text{for all }f\in\mathcal C\tag3.$$

In the context of the question, $S$ is the space $C([0,\infty),E)$ of càdlàg functions $[0,\infty)\to E$ equipped with the Skorohod metric. My hope is that we somehow can benefit from the fact that $X$ is not only càdlàg, but continuous.

Remark: I don't think that it's of any use, but let me remark that in my application $X^n$ and $X$ are Markov processes. Their transition semigroups, say $(\kappa_t)_{t\ge0}$ and $(\kappa^n_t)_{t\ge0}$, induce strongly continuous contraction semigroups, say $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$, on the space $C_0(E)$ of continuous functions $E\to\mathbb R$ vanishing at infinity equipped with the supremum norm. Each $(T_n(t))_{t\ge0}$ is even uniformly continuous. Moreover, $T_n(t)\xrightarrow{n\to\infty}T(t)$ with respect to the strong operator topology uniformly for $t$ in a bounded interval. If any of these properties are useful, feel free to use them.


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