Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to 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.