Heavy-tailed distribution

In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded:cite book |author=Asmussen, Søren |title=Applied probability and queues |publisher=Springer |location=Berlin |year=2003 |isbn=9780387002118] 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.

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

:$lim\_\{x\; o\; infty\}\; e^\{lambda\; x\}Pr\; [X>x]\; =\; infty\; quad\; mbox\{for\; all\; \}\; lambda>0.,$

This is also written in terms of the tail distribution function $overline\{F\}(x)\; equiv\; Pr(X>x)$ as

:$lim\_\{x\; o\; infty\}\; e^\{lambda\; x\}overline\{F\}(x)\; =\; infty\; quad\; mbox\{for\; all\; \}\; lambda>0.,$

This is equivalent to the statement that the moment generating function of $F$, $M\_F(t)$, is infinite for all $t>0$ [Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999] .

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 if for all $t\; in\; mathbb\{R\}$

:$lim\_\{x\; o\; infty\}\; Pr\; [X>x+t|X>x]\; =1,$

or equivalently

:$overline\{F\}(x+t)\; sim\; overline\{F\}(x)\; quad\; mbox\{as\; \}\; x\; o\; infty.$

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.

ubexponential 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:

:$Pr(X\_1+X\_2\; leq\; x)\; =\; F^\{*2\}(x)\; =\; int\_\{-\; infty\}^\{infty\}\; F(x-y),dF(y).$

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

A distribution $F$ on the positive half-line is subexponential if

:$overline\{F^\{*2(x)\; sim\; 2overline\{F\}(x)\; quad\; mbox\{as\; \}\; x\; o\; infty.$This impliescite book |author=Embrechts, Paul; Claudia Klüppelberg; Mikosch, Thomas |title=Modelling Extremal Events for Insurance and Finance |publisher=Springer |location=Berlin |year=1997 |pages= |isbn=9783540609315] that, for any $n\; geq\; 1$,

:$overline\{F^\{*n(x)\; sim\; noverline\{F\}(x)\; quad\; mbox\{as\; \}\; x\; o\; infty.$

The probabilistic interpretation of this is that, for a sum of $n$ independent random variables $X\_1,ldots,X\_n$ withcommon distribution $F$,

:$Pr(X\_1+\; cdots\; X\_n>x)\; sim\; Pr(max(X\_1,\; ldots,X\_n)>x)\; quad\; mbox\{as\; \}\; x\; o\; infty.$

This is often known as the principle of the single big jump [Foss, Konstantopolous, Zachary, "Discrete and continuous time modulated random walks with heavy-tailed increments", Journal of Theoretical Probability, 20 (2007), No.3, 581—612] .

A distribution $F$ on the whole real line is subexponential if the distribution$F\; I(\; [0,infty))$ is [Willekens, E. Subexponentiality on the real line. Technical Report, K.U. Leuven(1986)] . Here $I(\; [0,infty))$ is the indicator functionof the positive half-line. Alternatively, a random variable $X$ supported on the real line is subexponential if and only if $X^+\; =\; max(0,X)$ 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.

Those that are one-tailed include:
*the Pareto distribution;
*the Log-normal distribution;
*the Weibull distribution with shape parameter less than 1;
*the Burr distribution;
*the Log-gamma distribution.

Those that are two-tailed include:
*The Cauchy distribution, itself a special case of
*the t-distribution;
*all of the Stable Distribution family, excepting the special case of the normal distribution within that family. Stable distributions may be symmetric or not.

ee also

*Fat tail
*Leptokurtic
*The Long Tail

References