This article is cited in 11 papers

Probability theory and mathematical statistics

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

A. A. Mogulskii<sup>ab

a</sup> Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia
^b Sobolev Institute of Mathematics, 4, pr. Koptyuga, 630090, Novosibirsk, Russia

Abstract: 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.

Keywords: 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.

UDC: 519.21

MSC: 60K05, 60F10

Received July 10, 2018, published January 24, 2019

DOI: 10.33048/semi.2019.16.002

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