Let $X\_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X\_i]$ and $S\_n=\frac{1}{n}\sum\_{i=1}^nX\_i$. This question concerns the tail probability $\Pr[S\_n \ge \mu+\epsilon]$, where $\epsilon>0$ is a constant.


In the case that $|X\_i|$ is bounded with probability one, Hoeffding's Inequality states that the tail probability decays to exponentially fast in $n$ for any $\epsilon > 0$. [EDIT: The paper <http://arxiv.org/pdf/1209.6396v1.pdf> (Theorem 1.1) used to state a theorem which made it seem that finite variance was sufficient, but this is not true, and the paper has now been edited to state the sufficient condition more precisely].


A more general sufficient condition is that each variable is sub-Gaussian - see Exercise 6 of <http://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/>


With no assumptions at all on $X\_i$, the probability may fail to decay exponentially. For example, under the Cauchy distribution (which has no first or second moment), the probability is bounded away from zero. However, is there a sufficient condition which is less restrictive than that of boundedness?


This question is related to Large Deviations Theory and the Chernoff bound. The following upper bound is straightforward:


$$\Pr[S\_{n}\ge\mu+\epsilon]=\Pr[e^{\tau\sum\_{i=1}^{n}X\_{i}}\ge e^{\tau n(\mu+\epsilon)}]\le\frac{\mathbb{E}[e^{\tau\sum\_{i=1}^{n}X\_{i}}]}{e^{\tau n(\mu+\epsilon)}}=\frac{\mathbb{E}[e^{\tau X\_{1}}]^{n}}{e^{\tau n(\mu+\epsilon)}}$$


where $\tau$ is an arbitrary positive constant, and the inequality is Markov's inequality. For the optimal $\tau$, and under some technical assumptions, one can obtain a lower bound with the same exponential behavior (see "Large Deviations Techniques and Applications" by Dembo/Zeitouni).


Is there a simple sufficient condition for the above Chernoff bound to decay exponentially, which is stronger than boundedness or sub-Gaussian?


[EDITED SINCE ORIGINAL POST - The original answer of "no" was in response to the questions "Does a bounded absolute first moment suffice?". I certainly agree with this answer of no, but I am still interested in finding a condition as general possible]



Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations)
Note that Cramer's theorem applies in dimension 1 w/out any moment assumptions, but of course the rate function may vanish (for this version see theorem 2.2.3 in Dembo-Zeitouni). So the question is equivalent to the question ``when does the rate function vanish'' (the rate function is $\sup\_{\lambda\in R}
(\lambda x-\Lambda(\lambda))$, where $\Lambda(\lambda)= \log E(e^{\lambda X\_1}))$. So if 
$\Lambda(\lambda)=\infty$ for all $\lambda\neq 0$, clearly there is no exponential convergence, not just a failure of the upper bound.


