# I'm trying to understand a weak convergence result for Feller processes in Ethier and Kurtz

Let $$E$$ be a locally compact separable metric space, $$D([0,\infty),E)$$ denote the Skorohod space, $$C_0(E)$$ denote the space of continuous functions $$E\to\mathbb R$$ vanishing at infinity and $$E^\ast:=E\uplus\left\{\infty\right\}$$ denote the Alexandroff one-point compactification.

I'm trying to understand the proof of Theorem 2.5 of Chapter 4 in the book of Ethier and Kurtz: Let

• $$(T(t))_{t\ge0}$$ be a Feller (i.e. contractive, nonnegative and strongly continuous) semigroup on $$C_0(E)$$ with generator $$(\mathcal D(A),A)$$
• $$(T_n(t))_{t\ge0}$$ be a uniformly continuous semigroup on the space $$B(E)$$ of bounded measurable functions $$E\to\mathbb R$$ (equipped with the supremum norm) with (bounded) generator $$A_n$$
• $$D$$ be a core of $$(\mathcal D(A),A)$$
• $$(X_t)_{t\ge0}$$ and $$(X^n_t)_{t\ge0}$$ be $$E$$-valued càdlàg strong Markov processes on a probability space $$(\Omega,\mathcal A,\operatorname P)$$ with $$\operatorname E\left[f(X_{s+t})\mid\mathcal(X_r)_{r\in[0,\:s]}\right]=(T(t)f)(X_s)\;\;\;\text{almost surely for all }f\in C_0(E)\tag1$$ and $$\operatorname E\left[f(X^n_{s+t})\mid\mathcal(X^n_r)_{r\in[0,\:s]}\right]=(T_n(t)f)(X^n_s)\;\;\;\text{almost surely for all }f\in B(E)\tag2$$ for all $$s,t\ge0$$ and $$n\in\mathbb N$$

Assume $$\left\|T_n(t)f-T(t)f\right\|_\infty\xrightarrow{n\to\infty}0\;\;\;\text{for all }f\in C_0(E)\text{ and }t\ge0\tag3,$$ which is known to be equivalent to the claim that for all $$f\in D$$, there is a $$f_n\in B(E)$$ for all $$n\in\mathbb N$$ with $$\left\|f_n-f\right\|_\infty+\left\|A_nf_n-Af\right\|_\infty\xrightarrow{n\to\infty}0.\tag4$$ Moreover, assume $$X_0^n\to X_0\tag5$$ in distribution.

I want to show that $$(X^n_\ast\operatorname P)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),E))$$ is relatively compact, where $$\mathcal M_1(D([0,\infty),E))$$ denotes the space of probability measures on $$\mathcal B(D([0,\infty),E))$$ equipped with the topology of weak convergence$$^1$$.

The proof in Ethier and Kurtz utilizes quite general results. I would like to proof the claim directly without using these general results. The guideline seems to be to firstly show that $$(X^n_\ast\operatorname P)_{n\in\mathbb N}$$ is relatively compact when considered as a sequence in $$\mathcal M_1(D([0,\infty),E^\ast))$$.

How can we do that?

EDIT: For example, in the used Theorem 9.4 of Chapter 3, we should be able to choose $$C_a=C_0(E)$$ and $$D=\mathcal D(A)$$, noting that $$f(X^n_t)-\int_0^t(A_nf)(X^n_s)\:{\rm d}s\tag6$$ is a martingale wrt the filtration $$\mathcal F^{X_n}$$ generated by $$X^n$$, and hence the conditions $$(9.17)$$ and $$(9.18)$$ should be trivially satisfied. This should allow a much simpler proof of that Theorem.