Chernoff bound with an upper bound on the expectation
Suppose that $X$ is a sum of i.i.d Bernoulli random variables and $πΈ[π]=\mu$, according to the Multiplicative Chernoff bound we have that
$Pr(π \leq (1-\delta)\mu) \leq \exp (\frac{-\delta^2}{2}\mu)$, for all $0<\delta<1.$
My question: if we are given an upper bound on the expectation instead of a lower bound i.e. $\mu=E[X]\leq \mu'$, is it still true that there exists a constant π such that
$Pr(π \leq (1-\delta)\mu) \leq \exp (-c \delta^2 \mu')$ ?
Since it is easy to see if we are given a lower bound on the expectation i.e. $\mu' \leq E[X]=\mu$, then $Pr(π \leq (1-\delta)\mu) \leq \exp (\frac{-\delta^2}{2}\mu)\leq \exp (\frac{-\delta^2}{2}\mu')$, but if we just know an upper bound on $E[X]$ I tried to see it e.g. using stronger versions of Chernoff but couldn't ... and I want to check if it is true in the first case as I am not having any intuition ...
Thank you in advance.
$\endgroup$ 2 Reset to default
