TITLE: Moment generating function and exponentially decaying tails of probability distribution
QUESTION [5 upvotes]: This is a follow-up to this previous question.
Suppose I have a mean-zero symmetrically-distributed random variable $X$ over the support $\mathbb{R}$. If $X$ has a moment-generating function $M_X(t)$ that is smooth around 0, $X$ has an exponentially decaying tail probability, by Chernoff bound (Lemma 11.9.1 in Cover and Thomas's "Elements of Information Theory" 2nd edition).
Now, suppose that $X$ has an $M_X(t)$ that is not smooth around 0.  Suppose that $\mathbf{E}[X^k]=\infty$ for all even $k>n$, where $n$ is a positive integer.  Is there $X$ that has exponentially-decaying tail probability in that case?  Or would the tail probability always be a power-law?
Also, what happens to the tail if $M_X(t)$ is not defined, i.e. the integral in the transform diverges?
EDITS: Clarified the question based on helpful comments from @cardinal.

REPLY [8 votes]: I assume that "exponentially decaying tail probability" means that $P(|X| > t) \le C e^{-\epsilon t}$ for some $C, \epsilon$.  Any such random variable has finite moments of all orders.  This follows from the formula
$$E[|X|^p] = \int_0^\infty p t^{p-1} P(|X| > t) dt$$
which you can prove with Fubini's theorem and the fundamental theorem of calculus.