Large deviations for sums of independent random variables with dominatedly varying tails

Abstract

In this paper the large deviation results for partial and random sums

$$S_n - ES_n = \sum\limits_{i = 1}^n {X_i } - \sum\limits_{i = 1}^n {EX_i ,n \geqslant 1;S(t) - ES(t) = \sum\limits_{i = 1}^{N(t)} {X_i - E} \left( {\sum\limits_{i = 1}^{N(t)} {X_i } } \right)} ,t \geqslant 0$$

are proved, where {N(t); t ≥ 0} is a counting process of non-negative integer-valued random variables, and {X _n ; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.