I'm working on a problem that requires me to find an upper bound on the probability that the sum of independent draws from a random variable deviates from the expected value of that sum by more than a given constant. Specifically, let $X$ be a random variable and suppose that we draw $m$ values for $X$. Let $S$ be the sum of those draws: $S=\sum_{i=1}^m X_i$, where $X_i$ is the $i$-th draw from $X$. This sum has expected value $E[S]$. If we assume that $X$'s values are always in the interval of $[a, b]$, one could try to find an upper bound on the probability that the sum of the draws deviates from the expected value of the sum by more than $t$:
$P(S - E[S] > t)$
Hoeffding's inequality tells us that an upper bound for this probability is
$\exp\bigg( \frac{-2t^2}{\sum_{i=1}^m (a - b)^2} \bigg)$
The problem that I have requires me to find an upper bound on the probability that $k$ times the sum of draws deviates from the expected value by more than $t$:
$P(kS - E[S] > t)$
where $k$ is a constant.
It seems that it should be easy to find an upper bound for this probability, but I'm kind of stuck: I've tried simple algebraic manipulations in order to try to get rid of the $k$ an transform that probability into something that would allow me to use Hoeffding's bound; I also took a look at other bounds, like the Bernstein inequalities, but nothing seems quite right.
Does anybody have an idea for a bound for this type of probability? I feel that the answer is right in front of me but I can't see it...
Thanks in advance!