I have a power law distribution $X$ with exponent $c$: $$p(X=t) = \left\\{\begin{array}{cl}(c-1)/t^{c} & t \geq 1 \\\\ 0 & t < 1\end{array}\right.$$

From $X$ I take $n$ independent samples $X_1, \ldots, X_n$ and would like to give a tail bound for the mean; i.e. a good bound on $\Pr[\frac{1}{n}\sum X_i > t]$.

Unfortunately, my $X$ is unbounded and much of the work on concentration of measure assumes bounded variables. However, I do know that $X$ has bounded moments, so I can bound the variance and get a weak concentration. But if $c$ is larger, say $10$, I ought to be able to use the higher moments to get a stronger result. In particular it seems like the moment method outlined in http://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/ should be applicable, but the treatment there assumes bounded variables.

I can work through the modifications to that treatment myself and get a reasonable result, but it's kind of messy. So my question is: is there any existing published theorem that does what I want, so I don't have to add an ugly proof to my paper's appendix?