On exact large deviation principles for compound renewal processes

We consider two large deviation principles (LDPs): the “ordinary” LDP (when the “strong” Cramér condition is met) and the “extended” LDP when only the standard Cramér condition is met and the deviation functional may be finite also for discontinuous trajectories. The standard formulation of these principles involves two asymptotic (upper and lower) estimates for the logarithms of the probabilities that the normalized trajectory of the process lies in a given set $B$. We obtain conditions on a set $B$ such that these estimates coincide and the large deviation principles take the form of exact asymptotic equalities. Such LDPs are called exact. We show that the estimating interval of an ordinary LDP is contained in the estimating interval of the extended LDP. Hence the fulfillment of the exact extended LDP implies that of the exact ordinary LDP. The results obtained in the present paper are also fully valid and relevant for random walks (a special case of compound recovery processes).