considers the corresponding empirical measure process over $[0,T]$. Since at most one particle can jump at a given point of time, the measure $p_{\nu_N}^{(N)}$ is absolutely continuous with the measure corresponding to the above non-interacting system on $D([0,T],M_1(\mathcal{Z}))$. One can then write the Radon-Nikodym derivative using the Girsanov formula and show continuity properties of the same. An application of an extension of Sanov’s theorem (see [[@dawson-gartner-87 Theorem 3.5]]{}) tells us that the non-interacting particle system obeys the LDP on $D([0,T],M_1(\mathcal{Z}))$. The above theorem then follows by an application of Varadhan’s integral lemma (see [[@dembo-zeitouni Theorem 4.3.1]]{}). This approach has been carried out for a system of interacting diffusions in [@dawson-gartner-87] and for jump processes in [@leonard-95; @borkar-sundaresan-12].

We now recall a theorem that gives the large deviation principle for the sequence $\{p_{\nu_N,T}^{(N)}\}_{N \geq 1}$ on $M_1(\mathcal{Z})$. This can be obtained from the above theorem by an application of the contraction principle to the coordinate projection map $D([0,T], M_1(\mathcal{Z})) \ni \mu \mapsto \mu(T)$ (see [@dembo-zeitouni Theorem 4.2.1], [@borkar-sundaresan-12 Theorem 3.3]).

Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$. Then the sequence of probability measures $\{p_{\nu_N,T}^{(N)}\}_{N \geq 1}$ on the space $M_1(\mathcal{Z})$ satisfies the LDP with the good rate function 