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In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables:

http://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/

He then indicates that by weakening the tail hypothesis to $\operatorname{Pr}(|X|\geq t)\leq C\operatorname{exp}(-ct^p)$ with $0 < p \leq 1$, you can get a similar conclusion but with $\epsilon$ loss in the exponent of $n$. For my research, I am particularly interested in the case where $p=1/2$.

I have two main questions:

(1) Is there a reference that I can cite (and read) for the $0 < p \leq 1$ case?

(2) Is it known whether the $\epsilon$ loss is necessary?

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Related: mathoverflow.net/questions/118562 –  cardinal May 11 '13 at 22:32

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