Suppose that $X\_1, X\_2, \ldots, X\_n$ are i.i.d random variables distributed according to [Weibull distribution](http://en.wikipedia.org/wiki/Weibull_distribution) with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X\_i \geq t] = e^{-\Theta(t^{\epsilon})}$).


Now consider the random variable $S\_n = X\_1 + X\_2 + \ldots + X\_n$, when $n$ tends to infinity.
Clearly, $\mathbf{E}[S\_n] = O\_{\epsilon}(n)$. Is it true that for some $C = C(\epsilon)$ we have $\mathbf{Pr}[S\_n \geq C n] \leq e^{-\Omega\_{\epsilon}(n^{\alpha})}$ for some $\alpha = \alpha(\epsilon) > 0$? If so, what is the largest $\alpha$ one can get?


The standard MGF-based methods that work nicely in similar situations are not applicable here due to the fact that $X\_i$'s are heavy-tailed. My feeling is that this question must be studied somewhere.



Tails of sums of Weibull random variables
The affirmative answer can be found in [this](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=1116&option_lang=eng) paper by A.V.Nagaev: essentially "the conjecture" is true for $\alpha = \varepsilon$ (which is clearly the best possible).


