Local theorems for finite – dimensional increments of compound multidimensional arithmetic renewal processes with light tails

We continue to study the compound renewal processes under the Cramèr moment condition, which was started by A.A. Borovkov and A.A. Mogulskii (2013). In the present paper we study arithmetic multidimensional compound renewal process, for which the "control – ling" random vector $\xi=(\tau,\zeta)$ ($\tau>0$ determines the distance between the jumps, $\zeta$ determines the value of jumps of the compound renewal process) has an arithmetic distribution with light tails. For these processes we propose wide conditions (close to necessary), under which we can find exact asymptotics in local limit theorems for finite – dimensional increments.