Geodesic
Large deviations for sums of lattice random variables under the Cramer condition
Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 115-130.

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The sums of independent identically distributed random variables having a lattice distribution are considered. It is assumed that the unilateral Cramer condition holds in a bounded interval $(0,\lambda)$, that is, the extreme right conjugate distribution does not exist. Under an additional assumption on the regularity of the right tail of the underlying distribution, the local and integral theorems on large deviations of an arbitrarily high order are established. 
@article{DM_1998_10_3_a9,
     author = {A. V. Nagaev},
     title = {Large deviations for sums of lattice random variables under the {Cramer} condition},
     journal = {Diskretnaya Matematika},
     pages = {115--130},
     publisher = {mathdoc},
     volume = {10},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1998_10_3_a9/}
}
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A. V. Nagaev. Large deviations for sums of lattice random variables under the Cramer condition. Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 115-130. http://geodesic.mathdoc.fr/item/DM_1998_10_3_a9/