Large deviation principles for renewal–reward processes

We establish a sharp large deviation principle for renewal–reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cramér's theory. We also exhibit sufficient conditions for exponential tightness of renewal–reward processes, which leads to a full large deviation principle.