Large deviations for sums of lattice random variables under the Cramer condition

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.