Geodesic
Local theorems for arithmetic multidimensional compound renewal processes under Cramér's condition
Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 106-133 
A. A. Mogul'skiǐ; E. I. Prokopenko. Local theorems for arithmetic multidimensional compound renewal processes under Cramér's condition. Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 106-133. http://geodesic.mathdoc.fr/item/MT_2019_22_2_a6/
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     author = {A. A. Mogul'skiǐ and E. I. Prokopenko},
     title = {Local theorems for arithmetic multidimensional compound renewal processes under {Cram\'er's} condition},
     journal = {Matemati\v{c}eskie trudy},
     pages = {106--133},
     year = {2019},
     volume = {22},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2019_22_2_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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

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