Let $X_1, X_2, \ldots X_n$ be independent random variables, can we apply the Chernoff Bound Theorem to bound $E\left[\sum_i \Theta( \widehat{X_{i}}))\right]$ if $\Theta( \widehat{X_{i}}) : \widehat{X_{i}} \rightarrow \mathbb{R}$, i.e, $\Theta(\widehat{X_{i}})$ is a mapping from the random variable to a real number?
Thus, is the following Chernoff Bound applicable in the above scenario?$$P\left[\sum_i \Theta( \widehat{X_{i}})\leq(1-\epsilon) E\left[\sum_i \Theta( \widehat{X_{i}}))\right] \right] \leq e^{-\frac{\epsilon^2}{2} E\left[\sum_i \Theta( \widehat{X_{i}}))\right] }$$
Additionally, would the same Chernoff Bound be applicable for $E\left[\sum_i a_i \cdot \Theta( \widehat{X_{i}}))\right]$, where $a_i$ is a real numbered constant from the set $A$?
