Tail asymptotics of light-tailed Weibull-like sums

We consider sums of $n$ i.i.d. random variables with tails close to $\exp \{-x^{\beta }\}$ for some $\beta >1$. Asymptotics developed by Rootz'en (1987) and Balkema, Kl"uppelberg & Resnick (1993) are discussed from the point of view of tails rather of densities, using a somewhat different angle, and supplemented with bounds, results on a random number $N$ of terms, and simulation algorithms.