Let $(X_i)_{i \geq 1}$ be i.i.d. random variables such that $P(X_1 \geq t) \leq c_1e^{-c_2 t^{\beta}}$ for some $\beta > 0$. Then clearly $\mathbb{E}[X_i] < \infty$ and the strong law of large numbers tells us that $S_n = \frac{1}{n} \sum_{i=1}^{n} X_i$ converges to $\mathbb{E}[X_1]$ almost surely.
I want to obtain large deviations for $S_n$. In particular, I want an estimate of the form $P(S_n > n\mathbb{E}[X_1] + t) \leq k_1 e^{-k_2 t^{\alpha}}$.
I know that if $\beta \geq 2$, then $X_i$ are sub-gaussian and the tail is true with $\alpha=2$. However, I have not been able to find anything about the case when $\beta \in (0,2)$. I would appreciate any help.
These distributions were called "Sub-Weibull" in a recent paper, which proves concentration results of the flavor you are interested in.