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

Abstract

We continue the study of compound renewal processes (c.r.p.) under Cramer’s moment condition initiated in [2, 3, 6, 7, 8, 4, 5, 9, 10, 12, 16, 13, 14, 15]. We examine two types of arithmetic multidimensional c.r.p. $${\mathbf {Z}}(n)$$ and $${\mathbf {Y}}(n) $$ , for which the random vector $${\xi }=({\tau },{\boldsymbol {\zeta }})$$ controlling these processes ( $$\tau >0 $$ defines the distance between jumps, $$\boldsymbol {\zeta }$$ defines the value of jumps of the c.r.p.) has an arithmetic distribution and satisfies Cramer’s moment condition. For these processes, we find the exact asymptotics in the local limit theorems for the probabilities $$ {\mathbb {P}}\left ({{\mathbf {Z}}(n)={\boldsymbol {x}}}\right ), \quad {\mathbb {P}}\left ({{\mathbf {Y}}(n)={\boldsymbol {x}}}\right )$$ in the Cramer zone of deviations for $${\boldsymbol {x}}\in {\mathbb Z}^d $$ (in [9, 10, 13, 14, 15], the analogous problem was solved for nonlattice c.r.p., where the vector $$ {\boldsymbol {\xi }}=(\tau ,{\boldsymbol {\zeta }}) $$ has a nonlattice distribution).