Convergence of large deviation rates based on a link between wave governed random motions and ruin processes

It is possible to consider a sequence of wave governed random motions such that their paths converge to the path of a suitable ruin process with exponentially distributed claim sizes (see the proof of Theorem 1 in [Mazza, C., Rulliére, D., 2004. A link between wave governed random motions and ruin processes. Insurance Math. Econom. 35, 205-222]). In this paper we verify that some large deviation rates for these wave governed random motions converge to the analogous rates for the same ruin process.