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.

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    $\begingroup$ I think much will depend on $c_n$, and also on $p$. It is a bit too much to try to cover all possible scenarios. $\endgroup$ – Iosif Pinelis Jul 31 at 12:02
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    $\begingroup$ @losif Pinelis, I agree. $p$ is a fixed number in $(0,1)$. I just wanted to know if there is a common way to derive concentration inequalities of this type... $\endgroup$ – Somabha Jul 31 at 13:37
  • $\begingroup$ Please give a specific choice of $c_n$ that you care about. In particular if $c_n$ is a constant $>0$ then concentration is not around the mean but around the geometric mean. $\endgroup$ – Yuval Peres Jul 31 at 21:41

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