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 $$\underset{n\to \mathrm{\infty }}{\lim }\underset{s\ge t_n}{\sup }\mid \frac{P[S_n>s]}{n(1-F(s))}-1\mid =0$$ 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 $$\underset{t\to \mathrm{\infty }}{\lim }\frac{1-F(t+x)}{1-F(t)}=1\text{for each real }x\text{.}$$