I'm working on a problem where I need information on the size of $E_n=|S_n-n\mu|$, where $S_n=X_1+\ldots+X_n$ is a sum of i.i.d. random variables and $\mu=\mathbb EX_1$. For this to make sense, the $(X_i)$ have to be integrable. In that case, the weak law of large numbers says $E_n/n$ converges to 0 in probability, while the strong law says $E_n=o(n)$ almost surely.

If $X_1$ is square integrable, then we get the (stronger) result $E_n/(n^{1/2+\epsilon})$ converges to 0 in probablility. What I am looking for is a result for the case $\mathbb E|X_1|^\alpha<\infty$ for some $1<\alpha<2$. Back-of-the-envelope calculations suggest a result of the form $E_n=O(n^{1/\alpha+\epsilon})$ holds. I suspect a relationship with $\alpha$-stable laws and have tracked down Gnedenko and Kolmogorov's book, but find (1) their notation is quite hard to read; (2) they seem to mainly care about something that's not so important for me; namely a result of the Central Limit Theorem type. They impose extra conditions under which suitably scaled and translated versions of $S_n$ converge to a non-trivial distribution. I don't want to assume anything beyond an $\alpha$th moment inequality, but am looking for a correspondingly weaker conclusion giving upper bounds on $E_n$.

Can anyone point me to some results about deviation from the mean for sums of iid random variables with $\alpha$th moments ($1<\alpha<2$)?

In a similar vein, what if the random variables fail to have an $\alpha$th moment for some $\alpha<1$. Here I'd expect to see some kind of result telling me that if $S_n/a_n$ converges in probability to a constant for some sequence $(a_n)$, then that constant must be 0.

**Edit:**
Let me give a precise Chebyshev-like statement I would really like to be true.
Let $(X_n)$ be iid; $\mathbb EX_1=0$ and $\mathbb E|X_1|^\alpha<\infty$ for some $1<\alpha<2$.
Then $\mathbb P(|S_n|>t)<C_\alpha n\mathbb E|X_1|^\alpha/t^\alpha$, where $C_\alpha$ is a constant that only depends on $\alpha$ (I have guessed the bound based on "symbol-bashing" with the $\alpha$-stable law and believe it's at least "dimensionally correct".)

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