# Definition of Tail-index of a probability distribution

What is a valid definition of Tail-index of a probability distribution?

I understand that it is something to do with the rate of convergence of the density function $f(x)$ $($to $0)$ as $x \to \infty$. I tried searching google and I do find a lot of articles/papers on the topic but nowhere I can find a specific definition of the term.

Any help would be much appreciated! Thanks.

I have found a defintion at http://freakonometrics.hypotheses.org/2338, although again it is not presented as a primary definition of “tail index” but rather as an adjunct to the discussion of what they call heavy-tailed distributions. So I have had to fill in some parts to add a bit of rigor, and other people should check these parts.

The parts obtained from the freakonometrics article are in the shaded areas.

Consider any distribution $P(X)$ with cumulative distribution function $F(x) = 1- \overline{F}(x)$ defined by $\mbox{Pr }(X > x) = \overline{F}(x)$, such that
for some $\xi>0$,

where $\mathcal{L}(x)$ is some slowly varying function for large $x$.

The tail index of the fat-tailed distribution $P(X)$ is by definition $\xi$.

Although the freakomomics article calls this a heavy-tailed distribution, in Wikipedia and elsewhere this is called a fat-tailed distribution. The definition of a “slowly varying function” is that for all $a>0$

Thus we can restate the condition as $\overline{F}(x) \sim x^{-1/\xi}$ for some $\xi>0$, where $\sim$ denotes asymptotic equivalence.

The definition of a heavy-tailed distribution given in https://en.wikipedia.org/wiki/Heavy-tailed_distribution is that $P(X)$ is a heavy-tailed distribution if for all $\lambda > 0$,

All fat-tailed distributions are heavy-tailed in this sense, but not vice-versa.

For example, consider the probability distribution function For any positive $\lambda$
so the distribution is heavy-tailed. But for any positive $\xi$,

which implies that the distribution is not fat-tailed

Equivalently, there exists a slowly varying function $\mathcal{L}^*(x)$ such that for $0,

$\xi$ can by visualized as the opposite [negative] of the slope, at small $p$, of $\log F^{-1}(1-p)$ when that is plotted against $p$.

Somebody should add this definition to the Wikipedia page on heavy-tailed distributions, just above the section on Pickand's estimator of the tail index. However, I think the Freakonometrics reference is inadequate, both because it is not primarily intended as a definition of the term, and because the confusion about heavy-tailed and fat-tailed reduces confidence in using that as a reference.