Heavy-tailed distribution

In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:^[1] that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy.

There are two important subclasses of heavy-tailed distributions, the long-tailed distributions and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class.

There is still some discrepancy over the use of the term heavy-tailed. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a variance. The definition given in this article is the most general in use, and includes all distributions encompassed by the alternative definitions, as well as those distributions such as log-normal that possess all their power moments, yet which are generally acknowledged to be heavy-tailed. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)

1 Definition of heavy-tailed distribution
2 Definition of long-tailed distribution
3 Subexponential distributions
4 Common heavy-tailed distributions
5 See also
6 References

Definition of heavy-tailed distribution

The distribution of a random variable X with distribution function F is said to have a heavy right tail if^[1]

$\lim_{x \to \infty} e^{\lambda x}\Pr[X>x] = \infty \quad \mbox{for all } \lambda>0.\,$

This is also written in terms of the tail distribution function

$\overline{F}(x) \equiv \Pr(X>x) \,$

$\lim_{x \to \infty} e^{\lambda x}\overline{F}(x) = \infty \quad \mbox{for all } \lambda>0.\,$

This is equivalent to the statement that the moment generating function of F, M_F(t), is infinite for all t > 0^[2].

The definitions of heavy-tailed for left-tailed or two tailed distributions are similar.

Definition of long-tailed distribution

The distribution of a random variable X with distribution function F is said to have a long right tail^[1] if for all t > 0,

$\lim_{x \to \infty} \Pr[X>x%2Bt|X>x] =1, \,$

or equivalently

$\overline{F}(x%2Bt) \sim \overline{F}(x) \quad \mbox{as } x \to \infty. \,$

This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level: if you know the situation is bad, it is probably worse than you think.

All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.

Subexponential distributions

Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables $X_1,X_2$ with common distribution function $F$ the convolution of $F$ with itself, $F^{*2}$ is defined, using Lebesgue-Stieltjes integration, by:

$\Pr(X_1%2BX_2 \leq x) = F^{*2}(x) = \int_{- \infty}^{\infty} F(x-y)\,dF(y).$

The n-fold convolution $F^{*n}$ is defined in the same way. The tail distribution function $\overline{F}$ is defined as $\overline{F}(x) = 1-F(x)$ .

A distribution $F$ on the positive half-line is subexponential^[1] if

$\overline{F^{*2}}(x) \sim 2\overline{F}(x) \quad \mbox{as } x \to \infty.$

This implies^[3] that, for any $n \geq 1$ ,

$\overline{F^{*n}}(x) \sim n\overline{F}(x) \quad \mbox{as } x \to \infty.$

The probabilistic interpretation^[3] of this is that, for a sum of $n$ independent random variables $X_1,\ldots,X_n$ with common distribution $F$ ,

$\Pr(X_1%2B \cdots X_n>x) \sim \Pr(\max(X_1, \ldots,X_n)>x) \quad \mbox{as } x \to \infty.$

This is often known as the principle of the single big jump^[4].

A distribution $F$ on the whole real line is subexponential if the distribution $F I([0,\infty))$ is^[5]. Here $I([0,\infty))$ is the indicator function of the positive half-line. Alternatively, a random variable $X$ supported on the real line is subexponential if and only if $X^%2B = \max(0,X)$ is subexponential.

All subexponential distributions are long-tailed, but examples can be constructed of long-tailed distributions that are not subexponential.

Common heavy-tailed distributions

All commonly used heavy-tailed distributions are subexponential.^[3]

Those that are one-tailed include:

the Pareto distribution;
the Log-normal distribution;
the Lévy distribution;
the Weibull distribution with shape parameter less than 1;
the Burr distribution;
the Log-gamma distribution;
the Log-Cauchy distribution, sometimes described as having a "super-heavy tail" because it exhibits logarithmic decay producing a heavier tail than the Pareto distribution.^[6]^[7]

Those that are two-tailed include:

The Cauchy distribution, itself a special case of both the stable distribution and the t-distribution;
The family of stable distributions,^[8] excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. Lévy distribution. See also financial models with long-tailed distributions and volatility clustering.
The t-distribution.
The skew lognormal cascade distribution.^[9]

References

^ ^a ^b ^c ^d Asmussen, Søren (2003). Applied probability and queues. Berlin: Springer. ISBN 9780387002118.
^ Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999
^ ^a ^b ^c Embrechts, Paul; Claudia Klüppelberg; Mikosch, Thomas (1997). Modelling Extremal Events for Insurance and Finance. Berlin: Springer. ISBN 9783540609315.
^ Foss, Konstantopolous, Zachary, "Discrete and continuous time modulated random walks with heavy-tailed increments", Journal of Theoretical Probability, 20 (2007), No.3, 581—612
^ Willekens, E. Subexponentiality on the real line. Technical Report, K.U. Leuven(1986)
^ Falk, M., Hüsler, J. & Reiss, R. (2010). "Laws of Small Numbers: Extremes and Rare Events". Springer. p. 80. ISBN 9783034800082.
^ Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions". http://docentes.deio.fc.ul.pt/fragaalves/SuperHeavy.pdf.
^ John P. Nolan (2009). "Stable Distributions: Models for Heavy Tailed Data" (PDF). http://academic2.american.edu/~jpnolan/stable/chap1.pdf. Retrieved 2009-02-21.
^ Stephen Lihn (2009). "Skew Lognormal Cascade Distribution". http://www.skew-lognormal-cascade-distribution.org/.