Estimator for sum of independent and identically distributed (iid) variables
You need McDiarmid's inequality, or in particular the Hoeffding inequality, which is a simplification: 

Given $n$ independent $X_i$ with the property that $X_i \in [a_i, b_i]$, and $X = (1/n)\sum X_i$, then 

$$ \text{Pr}(|X - EX| \ge t) \le 2 \exp(\frac{2t^2n^2}{\sum (b_i - a_i)^2}) $$