I have a positive random variable $X$ with $E[X] = 1$ and a small number $k$ more moments bounded by constants:

$$E[(X-1)^i] = O(1) \forall i \in \{2, ..., k\}.$$

I'd like to bound the average of $n$ independent samples of $X$. Markov's inequality only uses the first moment to get:

$$\Pr[\frac{1}{n}\sum_{i=1}^n x_i \geq c] \leq 1/c$$

Chebyshev's inequality also uses the second moment to get:

$$\Pr[\frac{1}{n}\sum x_i \geq 1 + c] \leq \frac{O(1)}{nc^2}$$

which is better asymptotic behavior in $n$. If infinite moments converged, I could use Azuma/Chernoff/Hoeffding/McDiarmid to get

$$\Pr[\frac{1}{n}\sum x_i \geq 1 + c] \leq e^{-f(c)n}$$

for some function $f(c)$. But what if I have somewhere between 2 and infinite moments, say 10 moments. Is there a theorem with better asymptotic behavior for intermediate moments?