Some limit theorems for large deviations of sums of independent random variables with a common distribution function from the domain of attraction of the normal law

Abstract

We investigate some aspects of asymptotic behavior of the probabilities P (Sn ≥ α bn), where Sn is a sum of n independent random variables with common distribution function from the domain of attraction of the normal law, α is a positive number, and bn tends to infinity. In particular, necessary and sufficient conditions are given under which the series \(\sum\limits_n {fnP\left( {Sn \geqslant \alpha bn} \right)} \) converges or, after some normalization, has a limit as α ↘ α0, where α0 is a positive constant and fn is some positive sequence of a rather general form. Bibliography: 14 titles.