Local Theorems for Arithmetic Multidimensional Compound Renewal Processes under Cramér’s Condition

We continue the study of compound renewal processes (c.r.p.) under Cramér’smoment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensionalc.r.p. Z(n) and Y(n), for which the random vector ξ = (τ, ζ) controlling these processes (τ > 0 defines the distance between jumps, ζ defines the value of jumps of the c.r.p.)has an arithmetic distribution and satisfies Cramér’s moment condition. For theseprocesses, we find the exact asymptotics in the local limit theorems for the probabilities P (Z(n) = x), P (Y(n) = x) in theCramér zone of deviations for x ∈ Z^d (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p.,where the vector ξ = (τ, ζ) has a nonlattice distribution).