This article is cited in 11 papers

Probability theory and mathematical statistics

Integro-local theorems for multidimensional compound renewal processes, when Cramer's condition holds. I

A. A. Mogulskii^ab, E. I. Prokopenko<sup>ab

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

Abstract: In the work, which consists of 4 papers (the article and [15]–[17]), we obtain integro-local limit theorems in the phase space for multidimensional compound renewal processes, when Cramer's condition holds.
In the part I (the article) we consider the so-called first renewal process $\mathbf{Z}(t)$ in a regular region, which is an of analog Cramer's deviation region for random walk. The regular region includes normal and moderate deviations.

Keywords: compound multidimensional renewal process, first (second) renewal process, large deviations, integro-local limit theorems, renewal measure, Cramer's condition, deviation (rate) function, second deviation (rate) function.

UDC: 519.21

MSC: 60K05, 60F10

Received February 5, 2018, published May 4, 2018

DOI: 10.17377/semi.2018.15.041

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