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

Abstract

We continue the study of the compound renewal 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 ξ = (τ,ζ) has the arithmetic distribution, where τ > 0 defines the distance between jumps, ζ defines the values of jumps. For this processes the fine asymptotics in the local limit theorem for probabilities P(Z(n) = x) has been obtained in Cramer's deviation region of x $\in$ Z. In [6]-[10] the similar problem has been solved for non-lattice c.r.p., when the vector ξ = (τ,ζ) has the non-lattice distribution.