Abstract: This paper is a further investigation of large deviations for sums of random variables Sn=sum form i=1 to n Xi and S(t)=sum form i=1 to N(t) Xi,(t≥0), where {X_n,n≥1) are independent identically distribution and non-negative random variables, and {N(t),t≥0} is a counting process of non-negative integer-valued random variables, independent of {X_n,n≥1}. In this paper, under the suppose F∈G, which is a bigger heavy-tailed class than C, proved large deviation results for sums of random variables. 

Key words: large deviation;heavy-tailed distribution;strongly subexponential distribution, lognormal distribution