Integro-local limit theorems for compound renewal processes under Cramér's condition. I

We obtain integro-local limit theorems in the phase space for compound renewal processes under Cramér's moment condition. These theorems apply in a domain analogous to Cramér's zone of deviations for random walks. It includes the zone of normal and moderately large deviations. Under the same conditions we establish some integro-local theorems for finite-dimensional distributions of compound renewal processes.