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.

For an academic reference, a definition is given in the paper "The impact of competition on prices with numerous firms" by Xavier Gabaix, David Laibson, Deyuan Li, Hongyi Li, Sidney Resnick and Casper G. de Vries (Journal of Economic Theory, 2016) -- see Definition 1. For any distribution with cdf $F$ and pdf $f$, and support $(w_{l}, w_{u})$, the tail index $\gamma$ is defined
$$ \gamma := \lim_{x \to w_{u}}\frac{d}{dx}\left(\frac{1-F(x)}{f(x)}\right).$$
They provide a number of examples in their Table 1.
