Fat Tails - Definition

Definition

The distribution of a random variable X is said to have a fat tail if

$\Pr \sim x^{- \alpha}\text{ as }x \to \infty,\qquad \alpha > 0.\,$

That is, if X has a probability density function, ,

$f_X(x) \sim x^{ - (1 + \alpha)} \text{ as }x \to \infty, \qquad \alpha > 0.\,$

Here the notation "" is the "twiddles" notation used for the asymptotic equivalence of functions. Some reserve the term "fat tail" for distributions only where 0 < α < 2 (i.e. only in cases with infinite variance).

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