results are known in the case of both Markov jump processes as well as diffusion processes; see [[@borkar-sundaresan-12 Appendix]]{}, and [[@freidlin-wentzell Chapter 6, Section 2]]{}. The main ingredients that are used in proving these results are (i) the strong Markov property of the $\mu_N$ process, (ii) Theorem \[thm:finite-duration-ldp\] and Corollary \[cor:uniform\_ldp\] on the LDP for finite time durations, and (iii) the joint continuity of the terminal time rate function $S_T(\cdot|\cdot)$ (see [[@borkar-sundaresan-12 Lemma 3.3]]{}). In the following results, $P_\nu$ denotes the solution to the $D([0,\infty),M_1(\mathcal{Z}))$-valued martingale problem for $L^N$ with initial condition $\nu \in M_1^N(\mathcal{Z})$ and $E_\nu$ denotes the corresponding expectation; note that in $P_\nu$ and $E_\nu$ we suppress the dependence on $N$ for ease of readability.

Let $K \subset M_1(\mathcal{Z})$ be a compact set such that all points in $K$ are equivalent to each other. Then, given $\varepsilon> 0$, there exist $\delta > 0$ and $N_0 \geq 1$ such that for all $N \geq N_0$ and $\nu \in  [K]_{\delta}\cap M^N_1(\mathcal{Z})$, $$\begin{aligned}
E_\nu \tau_{[K]_{\delta}} \leq \exp\{N\varepsilon\}.\end{aligned}$$ \[lemma:fw17\]

Let $K \subset M_1(\mathcal{Z})$ be a compact set and $G$ be a neighbourhood of $K$. Then, given $\varepsilon > 0$, there exist $\delta>0$ and $N_0 \geq 1$ such that for all $\nu \in 