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Local theorems for arithmetic multidimensional compound renewal processes under Cramér's condition
A. A. Mogul'skiĭa,
E. I. Prokopenkob a Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia
b Novosibirsk State University, Novosibirsk, 630090 Russia
Abstract:
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).
Key words:
compound renewal process, Cramér's condition, arithmetic distribution, renewal function, deviations function, large deviations, moderate large deviations, local limit theorem.
UDC:
519.214 Received: 04.02.2019
Revised: 08.05.2019
Accepted: 10.06.2019
DOI:
10.33048/mattrudy.2019.22.207
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