Say $X_1, \cdots, X_n$ are i.i.d random variables with mean zero, let $S_n = \sum_{i=1}^n X_i$, we know by SLLN $$\frac{S_n}{n}\rightarrow 0\text{ a.s}$$
We could further know that the sequence of random variables $\{\frac{S_n}{n}\}$ are uniformly integrable. Hence u.i + converge in prob(weaker than a.s) implies that $$\mathbb{E}\left(\left|\frac{S_n}{n}\right|\right)\rightarrow 0$$
For classical large deviation theory, such as Cramer's rule(though for r.vs that has moment generating function existed), and roughly speaking, it tells us the speed of the probability convergence, $$P\left(\frac{S_n}{n}\in A\right) \sim e^{-nI(A)}$$(Please forgive me not rigorous here).
So for the L1 convergence as I mentioned above, could we have a convergence speed statement? Such as $$\mathbb{E}\left(\left|\frac{S_n}{n}\right|\right)= O(\frac{1}{n})$$(I just pick one decreasing order as $O(\frac{1}{n})$, it definitely could be something else)
I would like to find the result of the above type. Could anyone give me a hint on what literature or paper should I look at?