Geodesic
Integro-local theorems for multidimensional compound renewal processes, when Cramer's condition holds. I
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 475-502

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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. 
@article{SEMR_2018_15_a35,
     author = {A. A. Mogulskii and E. I. Prokopenko},
     title = {Integro-local theorems for multidimensional compound renewal processes, when {Cramer's} condition holds. {I}},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {475--502},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a35/}
}
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A. A. Mogulskii; E. I. Prokopenko. Integro-local theorems for multidimensional compound renewal processes, when Cramer's condition holds. I. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 475-502. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a35/