According to Wikipedia the most extreme case of a fat tail follows a power law:

The most extreme case of a fat tail is given by a distribution whose tail decays like a power law.

That is, if the complementary cumulative distribution of a random variable X can be expressed as

$$Pr[X>x] \sim x^{-\alpha} \quad \text{as} \quad x \to \infty, \quad \alpha >0$$

For these cases we have that for some sample size of size at least $n$ there are order statistics that have a finite expectation value.

However, in relation to a question about infinite/finite expectation values of order statistics, I got to think of a special case of distributions for which there is no size $n$ such that the order statistic will have a finite expectation value. This occurs when the quantile function has an essential singularity.

An example is $$Q(p) = e^{1/(1-p)} - e$$ for which the distribution function is

$$F(x) = \begin{cases} 0 \quad &\text{if} &\quad x<0 \\ 1 - \frac{1}{\log(x+e)} \quad &\text{if} &\quad x\geq 0 \\ \end{cases}$$ or $$f(x) = \begin{cases} 0 \quad &\text{if} &\quad x<0 \\ \frac{1}{(x+e)\log(x+e)^2} \quad &\text{if} &\quad x\geq 0 \\ \end{cases}$$

another case is discussed here: https://stats.stackexchange.com/a/417418/164061 the distribution functions that approach a power law can be bounded above by a linear function on a log-log plot, functions that are not like that will have in some sense 'more fat' tails than a distribution function that approaches a power law.

So it seems that we can think of distributions that have even more extreme tails than $Pr[X>x] \sim x^{-\alpha}$

Are there descriptions of fat tailed distributions that have this property? For instance do they have a particular name? (I suggest ultra-fat tailed distribution, if none exists yet)

  • $\begingroup$ Apparently there is a concept of super-heavy-tailed and the log-Cauchy distribution is an example, but I do not know an original reference or where the term originates. $\endgroup$ – Sextus Empiricus May 29 at 10:30
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    $\begingroup$ Falk et al. (2011) devote a section to Super-Heavy Tail analysis (section 2.7) with a subsection called "Super-Heavy Tails in the Literature" and more. $\endgroup$ – COOLSerdash May 29 at 10:48
  • $\begingroup$ In this case, Wiki is misleading. Tails heavier than an exponential power law can be called super-exponential tails. In other words, they grow faster than an exponential tail. One way to identify them is in a log-log graph which linearizes exponential growth. In this case, super-exponential growth curves sharply away (either upward or downward) from the linearized exponential. The current Covid-19 pandemic is probably one example of such super-exponential growth, one description of which is here...en.wikipedia.org/wiki/Dragon_king_theory $\endgroup$ – user332577 May 29 at 12:49
  • $\begingroup$ @user332577 that is an interesting link. But I am not talking about super-exponential tails, I am talking about super-power-law tails, which are even more fat. $\endgroup$ – Sextus Empiricus May 29 at 13:28
  • $\begingroup$ You need to clarify the distinction. Why wouldn't they have the same mechanism? $\endgroup$ – user332577 May 29 at 14:43

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