# Can Chernoff Bound Theorem be applied to functions of independent random variables

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 $$A$$?

• Are you sure you quote Chernoff bound correctly? en.wikipedia.org/wiki/Chernoff_bound Commented Apr 15 at 8:33
• this is another version of the Chernoff Bound I found in lecture notes @vanderWolf Commented Apr 15 at 8:37