Tail distribution function:

\[ \overline{F}(x) \equiv \Pr[X>x] = 1 - F(x) \]

Heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:

\[ \forall \lambda>0,\quad \lim_{x \to \infty} e^{\lambda x} \overline{F}(x) = \infty \]

There are three important subclasses of heavy-tailed distributions:

  1. Fat-tailed distribution: \[ \exists \alpha, c > 0,\quad \lim_{x \to \infty} x^{\alpha} \overline{F}(x) = c \]
  2. Subexponential distribution: \[ \forall n \in \mathbb{Z}^+, \lim_{x \to \infty} \overline{F}_{X_{(n)}}(x) / \overline{F}_{\sum X_i}(x) = 1 \]
  3. Long-tailed distribution: \[ \forall t>0, \lim_{x \to \infty} \overline{F}(x+t) / \overline{F}(x) = 1 \]

All subexponential distributions are long-tailed.

Examples of heavy-tailed distributions:

  • Fat-tailed: log-logistic, Pareto;
  • Subexponential: ;
  • Long-tailed distribution: ;

log-normal, Weibull, Zipf, Cauchy, Student's t, Frechet

Resources

Jayakrishnan Nair, Adam Wierman, Bert Zwart. The Fundamentals of Heavy-Tails: Properties, Emergence, and Identification.

🏷 Category=Probability