The connection between a random variable's moments and its tails Suppose I know all the moments of some random variable XX. When is knowledge of the moments sufficient to give one an understanding of the tail behavior?
Is there a nice way to show that a random variable has exponential tails just by looking at its moments? In particular, I would be very curious if one could use merely the moments of the normal distribution to show that it has exponential tails. Is this possible?
 A: Moments can be a guide to tail behavior only for those distributions for which the moments are defined. Nassim Taleb's free book Silent Risk (downloadable from his website) has one of the best expositions of heuristics that leverage the moments in defining the tails. But, and this is an important but, there are entire classes of probabilistic distributions which have infinite or undefined moments, e.g., the Cauchy or Lévy to name two. 
The power exponent associated with the tail index offers one approach to classifying the behavior of a distribution that does not rely on the moment structure. While some contend that this tail exponent is readily estimable from a simple regression model, others are much less sanguine about this approach. E.g., see this paper by Clauset, Shalizi and Newman on tail exponents:  https://www.cs.purdue.edu/homes/agebreme/Networks/papers/clauset-powerLaw-siamRev05.pdf  To quote from the abstract:

Power-law distributions occur in many situations of scientific
  interest and have significant consequences for our understanding of
  natural and man-made phenomena. Unfortunately, the detection and
  characterization of power laws is complicated by the large
  fluctuations that occur in the tail of the distribution—the part of
  the distribution representing large but rare events— and by the
  difficulty of identifying the range over which power-law behavior
  holds. Commonly used methods for analyzing power-law data, such as
  least-squares fitting, can produce substantially inaccurate estimates
  of parameters for power-law distributions, and even in cases where
  such methods return accurate answers they are still unsatisfactory
  because they give no indication of whether the data obey a power law
  at all. Here we present a principled statistical framework for
  discerning and quantifying power-law behavior in empirical data. Our
  approach combines maximum-likelihood fitting methods with
  goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and
  likelihood ratios. We evaluate the effectiveness of the approach with
  tests on synthetic data and give critical comparisons to previous
  approaches. We also apply the proposed methods to twenty-four
  real-world data sets from a range of different disciplines, each of
  which has been conjectured to follow a power law distribution. In some
  cases we find these conjectures to be consistent with the data while
  in others the power law is ruled out.

In addition, this Wikipedia entry on the Tweedie distribution has an example of how the tail exponent can be used to assign distributions to one of several possible extreme value behaviors...
https://en.wikipedia.org/wiki/Tweedie_distribution
In the example you give using the normal distribution, its moments and, hence, its tails, are finite. In fact, since the normal possesses a finite moment structure, it's well known to be very "thin-" as opposed to "fat-" tailed. This is all very nicely explained in David Hand's excellent and recent book The Improbability Principle. Hand goes to great lengths to reinforce the notion that the normal is quite robust to violations, but this is definitely not the same thing as being "heavy-tailed." 
The great example Hand provides is related to stock market returns following the August 2007 drawdowns and a Goldman-Sachs CFO's Senate testimony about the several day phenomena of "25-standard-deviation events," based on the normal distribution. As Hand notes, there aren't enough days since the beginning of time within which a 25-standard-deviation event could occur. In this case, a much better model of stock returns is the Cauchy distribution where these extreme moves have odds of occurrence of roughly 1 in 100 (see pages 147++).
