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Let $S_n$ be the sum of independent random variables with distribution $F$.Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$  \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right|  =0 $$ are given. These conditions extend and strengthen a series of previousresults. Additionally, a connection with subexponential distributions isdemonstrated. That is, $F$ is subexponential if and only if the condition aboveholds for some $t_n$ and $$  \lim_{t\to\infty}{1-F(t+x)\over 1-F(t)} = 1  \quad\text{for each real $x$.}$$

Large Deviation Probabilities for Sums of Random Variables with Heavy or Subexponential Tails