Assume we have a sequence of i.i.d. RV's $X_i$ and we know $\mathbb{E}[X_1]=m<\infty$. What can we say about the behaviour of $P(S_n\geq nk)$ with $S_n:=\sum_{i=1}^nX_i$ and $k>m$?
Here is what I think to know, please correct me if I'm wrong.
I mean, obviously this probability goes to $0$ by the law of large numbers, but at which rate? If we had $\mathbb{E}[e^{\lambda X_1}]<\infty$ for all $\lambda$, we could apply Cramér's theorem and get exponential decay. On the other hand, if we knew $\mathbb{E}[e^{\lambda X_1}]=\infty$ for all $\lambda> 0$, then we could conclude that the decay is not exponentially fast (also by Cramér).
But are there any other tools to make a statement about the speed of convergence? What if I don't know whether the exponential moments exist or if they exist for some $\lambda$ but not for others? Are there any general techniques for those cases or does it just come down to calculating the rate function and getting the result "by hand"?