Local theorems for arithmetic compound renewal processes when Cramer's condition holds

We continue the study of the compound reneal processes (c.r.p.), where the moment Cramer's condition holds (see [1]–[10], where the study of c.r.p. was started). In the paper arithmetic c.r.p. $Z(n)$ are studied. In such processes random vector $\xi = (\tau,\zeta)$ has the arithmetic distribution, where $\tau >0 $ defines the distance between jumps, $\zeta$ defines the values of jumps. For this processes the fine asymptotics in the local limit theorem for probabilities $\mathbf{P}(Z(n)=x)$ has been obtained in Cramer's deviation region of $x\in \mathbb{Z}$. In [6]–[10] the similar problem has benn solved for non-lattice c.r.p., when the vector $\xi=(\tau,\zeta)$ has the non-lattice distribution.