A Renewal version of the Sanov theorem

Large deviations for the local time of a process $X_t$ are investigated, where $X_t=x_i$ for $t \in [S_{i-1},S_i[$ and $(x_j)$ are i.i.d.\ random variables on a Polish space, $S_j$ is the $j$-th arrival time of a renewal process depending on $(x_j)$. No moment conditions are assumed on the arrival times of the renewal process.