Geodesic
Local theorems for arithmetic compound renewal processes when Cramer's condition holds
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 21-41.

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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.
Mots-clés : обобщенный процесс восстановления, арифметический обобщенный процесс восстановления, функция (мера) восстановления, моментное условие Крамера; функция уклонений, вторая функция уклонений, большие уклонения; умеренные уклонения, локальная предельная теорема. 
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     author = {A. A. Mogulskii},
     title = {Local theorems for arithmetic compound renewal processes when {Cramer's} condition holds},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {21--41},
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     volume = {16},
     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a38/}
}
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A. A. Mogulskii. Local theorems for arithmetic compound renewal processes when Cramer's condition holds. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 21-41. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a38/

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