Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

Local limit theorems are obtained for superlarge deviations of sums $S(n)=\xi(1)+\dots+\xi(n)$ of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of $\xi$ has the form $\mathbb P(\xi=k)=e^{-k^\beta L(k)}$, where $\beta>2$, $k\in\mathbb Z$ ($\mathbb Z$ is the set of all integers), and $L(t)$ is a slowly varying function as $t\to\infty$ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities $\mathbb P\bigl(S(n)=k\bigr)$ as $k/n\to\infty$, complement the results on superlarge deviations in [1, 2].