Chernoff inequalities Chernoff inequalities are inequalities that express concentration around the expectation of a random variable $X=\sum_iX_i$ where the $X_i$ are i.i.d random variables
I have been encountering these inequalities at differenet contexts and for several distributions of the $X_i's$. Nevertheless whenever I try to understand the proof for it, it seems to me that the proof is just a sequence of tricks and I fail to get any insight.


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*Do you know of any insightful, perhaps using "higher math", way to view these inequalities? 

*For which distributions of the $X_i's$ can one expect to get a chernoff bound for $\sum_iX_i$?

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*It's a quantitative version of the central limit theorem, in addition to some bounds on tail probabilities the normal distribution. You can see that the normal distribution decays rapidly from the probability density function $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$.
This leaves the question of an intuitive explanation for the central limit theorem. If something like the CLT were true, you would expect the probability distribution of $\frac{1}{\sqrt{n}}(X_1+\dots,X_n)$ (where $X_1,\dots,X_n$ are i.i.d.r.v with mean zero and bounded, say) to converge to a probability distribution whose moment generating function satisfies $M_X^n(t/\sqrt{n})=M_X(t)$, which must be of the form $\log M_X(t)=Ct^2$, and therefore must be a normal distribution.

*It suffices that the variables are independent and bounded. An important generalisation is Azuma's inequality.
