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

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 previous results. Additionally, a connection with subexponential distributions is demonstrated. That is, $F$ is subexponential if and only if the condition above holds for some $t_n$ and $$\lim_{t\to\infty}{1-F(t+x)\over 1-F(t)} = 1 \quad\text{for each real $x$.}$$