# Heavy-tailed distribution

(Redirected from Heavy tail)

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.

## Definition of Heavy-tailed Distribution

The distribution of a random variable X with distribution function ${\displaystyle F}$ is said to have a heavy right tail if[2]

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

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

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

This is equivalent to the statement that the moment generating function of ${\displaystyle F}$, ${\displaystyle M_{F}(t)}$, is infinite for all ${\displaystyle t>0}$[3].

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 ${\displaystyle F}$ is said to have a long right tail[4] if for all ${\displaystyle t\in \mathbb {R} }$

${\displaystyle \lim _{x\to \infty }\Pr[X>x+t|X>x)=1,}$

or equivalently

${\displaystyle {\overline {F}}(x+t)\sim {\overline {F}}(x)\quad {\mbox{as }}x\to \infty .}$

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 ${\displaystyle X_{1},X_{2}}$ with common distribution function ${\displaystyle F}$ the convolution of ${\displaystyle F}$ with itself, ${\displaystyle F^{*2}}$ is defined by:

${\displaystyle \Pr(X_{1}+X_{2}\leq x)=F^{*2}(x)=\int _{-\infty }^{\infty }F(x-y)F(dy).}$

The n-fold convolution ${\displaystyle F^{*n}}$ is defined in the same way.

A distribution ${\displaystyle F}$ is subexponential[5] if

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

This implies[6] that, for any ${\displaystyle n\geq 1}$,

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

The probabilistic interpretation[7] of this is that for a sum of ${\displaystyle n}$ independent random variables ${\displaystyle X_{1},\ldots ,X_{n}}$

${\displaystyle \Pr(X_{1}+\cdots X_{n}>x)\sim \Pr(\max(X_{1},\ldots ,X_{n})>x)\quad {\mbox{as }}x\to \infty .}$

This is often known as the principle of the single big jump[8].

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.[9]

Those that are one-tailed include:

Those that are two-tailed include:

## References

1. Asmussen, Applied Probability and Queues, 2003
2. Asmussen, Applied Probability and Queues, 2003
3. Rolski, Schmidli, Scmidt, Teugels, Stochastic Processes for Insurance and Finance, 1999
4. Asmussen, Applied Probability and Queues, 2003
5. Asmussen, Applied Probability and Queues, 2003
6. Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997
7. Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997
8. Foss, Konstantopolous, Zachary, "Discrete and continuous time modulated random walks with heavy-tailed increments", Journal of Theoretical Probability, 20 (2007), No.3, 581—612
9. Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997