Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cramér condition

This paper studies the asymptotic behavior of a density of a sum of independent identically distributed random variables with a common absolutely continuous distribution satisfying the right-hand Cramér condition. We prove that for a definite class of such distributions the well-known asymptotic representations in local and integral limit theorems are valid in the case of large deviations of arbitrarily high order.