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The current Wikipedia definition is

The distribution of a random variable $X$ with distribution function $F$ is said to have a heavy (right) tail if the moment generating function of $F,$ $MF(t),$ is infinite for all $t > 0,$ i.e.

$$\int_{-\infty}^\infty e^{tx}dF(x)=\infty, \forall \;t>0.$$

This is consistent with this upvoted post.

The question is why is there a need or advantage in resorting to MGFs to define this concept?

I look forward to less detailed, surprising answers, but as a note-to-self, page 9 of this document is a solid explanation already. More tangential, this is interesting.

kjetil b halvorsen
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Antoni Parellada
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    It's not quite clear what you're fishing for, because the motivation for this definition of heavy-tailed is explicit: it greatly reduces the utility of the mgf. Are you perhaps trying to ask what value it might have for the mgf to be defined in a real neighborhood of zero? – whuber Oct 25 '17 at 21:08
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    @whuber I was reading [N. Taleb's Silent Risk math draft here](http://www.fooledbyrandomness.com/SilentRisk.pdf), and trying to make sense of the definition of heavy tails as in the OP, and I thought that besides the fact that power law distributions have no finite moments, there could possibly be some easy way of seeing how the multiplication of the pdf by $e^{tx}$ might be a way to express that the tail is heavier than that of exponential functions. I am probably not answering your question, but this is the background of my post. – Antoni Parellada Oct 25 '17 at 21:35
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    The background helps. The connection between power law distributions and moments is clear--you're balancing exponents in an integral. The role of the exponential is not so clear. There's a hierarchy of rates at which a tail may decay. After the sequence of power-law tails $x^{-1},x^{-2},\ldots$ comes any exponential $e^{-tx}$ for $t\gt 0$. However, plenty of growth rates fit in between the two sequences, such as $x^{-1}e^{-tx}$ (for any $t\gt0$), $e^{-tx}/\log(x)$, $e^{-\sqrt{t}}$, etc. I think deeper insight can be had by considering the characteristic function in a complex neighborhood of 0. – whuber Oct 25 '17 at 22:09
  • A comment about the given "definition" of heavy-tailedness. One can define anything however one wants for local use, but the general application of heavy-tailed distributions is to model outlier-prone processes. For that purpose, the stated "definition" of heavy tailedness is too limited. After all, one can have outlier-prone processes that are bounded, e.g., mix a U(-1,1) with a U(-100, 100), with mixing probabilities .99 and .01. – BigBendRegion Dec 17 '19 at 12:33
  • Unarguably *one* application of heavy-tailed distributions is to model outliers, but that's far from a "general application." That's why I find this answer to be misleading. – whuber Dec 17 '19 at 15:22
  • I said "outlier-prone *processes*," not outliers. What's misleading about that? Oh that's right, you don't like the term "outlier." Substitute "processes that produce occasional observations that are quite far from the main body of the data." Better? – BigBendRegion Dec 18 '19 at 00:22
  • I'm fine with "outlier," Peter. My comment tried to explain that your characterization of heavy tails is overly narrow. – whuber Dec 18 '19 at 14:26
  • You miss the point. Quite on the contrary, the typical (above) characterizations of "heavy tails" are much narrower than what I have in mind, which would include them. Infinite tails are needlessly restrictive to model processes that produce extremes. Take flood height, eg. The maximum height, by the laws of physics alone, has 0 probability of being above some value M. Yet the process obviously produces extremes, eg, "100 year floods." If your client wanted a model for this, and you gave her/him the above definition of heavy tails, you would be doing a disservice by excluding logical models. – BigBendRegion Dec 20 '19 at 22:54
  • I am sufficiently familiar with your other posts here (and some of your articles, Peter) not to miss the point. Because we are now discussing matters of mathematical definitions I find there's nothing left to discuss and so will bow out. – whuber Dec 23 '19 at 13:44

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