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Sib. Èlektron. Mat. Izv., 2019, Volume 16, Pages 21–41 (Mi semr1051)  

Probability theory and mathematical statistics

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

A. A. Mogulskiiab

a 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.

Funding Agency Grant Number
Russian Science Foundation 18-11-00129


DOI: https://doi.org/10.33048/semi.2019.16.002

Full text: PDF file (228 kB)
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Bibliographic databases:

UDC: 519.21
MSC: 60K05, 60F10
Received July 10, 2018, published January 24, 2019

Citation: A. A. Mogulskii, “Local theorems for arithmetic compound renewal processes when Cramer's condition holds”, Sib. Èlektron. Mat. Izv., 16 (2019), 21–41

Citation in format AMSBIB
\Bibitem{Mog19}
\by A.~A.~Mogulskii
\paper Local theorems for arithmetic compound renewal processes when Cramer's condition holds
\jour Sib. \`Elektron. Mat. Izv.
\yr 2019
\vol 16
\pages 21--41
\mathnet{http://mi.mathnet.ru/semr1051}
\crossref{https://doi.org/10.33048/semi.2019.16.002}


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