Necessity of Cramer's, Linnik's and Statulevic̆ius conditions for large deviation probabilities

In this paper the necessity of Cramer's, Linnik's and Statulevic̆ius conditions is considered in theorems for large deviations probabilities. It is shown also that Statulevic̆ius condition is a generalization of Cramer's and Linnik's conditions.