I want a concentration inequality for the exponential of a weighted sum of independent Bernoulli random variables around its mean, for one of my research works. I was wondering if there is a well known tool in the theory of concentration inequalities that deals with such a case.
To be precise, suppose that $X_1,\ldots, X_n$ are iid. Bernoulli with mean $p$, and suppose that $a_1,\ldots,a_n$ are fixed $\pm 1$ valued real numbers. Define: $$G_n := \exp\left(c_n\sum_{i=1}^n a_i X_i\right)~.$$ In the above expression, $c_n$ is some suitable scaling factor. I am looking at concentration inequalities that bound the upper tail probability: $$\mathbb{P}\left(G_n - \mathbb{E} G_n > t\right)$$ for some fixed real number $t$. I tried to apply McDiarmid's bounded difference inequality and the concentration inequality for self-bounding functions, but they are not quite working (I am encountering a kind of circularity in bounding the differences). In some sense, I feel that these inequalities work best when the quantity whose concentration we want to derive, is in the form of a sum, not in the form of the exponential of a sum, as is here. Is there a known way in the literature to deal with this problem? Any help will be greatly appreciated.
