A commonly used form of the Chernoff bound is: For independent random variables $X_1,\dots,X_n$ in $(0,1)$ with $X=\sum X_i$ having expected value $\mu$, $$\text{Pr}(X\geq (1+\delta)\mu)\leq e^{-\delta^2\mu/3}$$ for $\delta>0$. A similar bound holds for $\delta<0$.
How important is the assumption that $X_i$ are positive random variables? If we have instead $X_i$ being in $(-1,1)$, do we have similar results?