# Heavy-tailed distribution

(Redirected from Heavy tail)

In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: 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

$\lim _{x\to \infty }e^{\lambda x}\Pr[X>x]=\infty \quad {\mbox{for all }}\lambda >0.\,$ This is also written in terms of the tail distribution function ${\overline {F}}(x)\equiv \Pr(X>x)$ as

$\lim _{x\to \infty }e^{\lambda x}{\overline {F}}(x)=\infty \quad {\mbox{for all }}\lambda >0.\,$ This is equivalent to the statement that the moment generating function of $F$ , $M_{F}(t)$ , is infinite for all $t>0$ .

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 $t\in \mathbb {R}$ $\lim _{x\to \infty }\Pr[X>x+t|X>x)=1,$ or equivalently

${\overline {F}}(x+t)\sim {\overline {F}}(x)\quad {\mbox{as }}x\to \infty .$ 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 $X_{1},X_{2}$ with common distribution function $F$ the convolution of $F$ with itself, $F^{*2}$ is defined by:

$\Pr(X_{1}+X_{2}\leq x)=F^{*2}(x)=\int _{-\infty }^{\infty }F(x-y)F(dy).$ The n-fold convolution $F^{*n}$ is defined in the same way.

A distribution $F$ is subexponential if

${\overline {F^{*2}}}(x)\sim 2{\overline {F}}(x)\quad {\mbox{as }}x\to \infty .$ This implies that, for any $n\geq 1$ ,

${\overline {F^{*n}}}(x)\sim n{\overline {F}}(x)\quad {\mbox{as }}x\to \infty .$ The probabilistic interpretation of this is that for a sum of $n$ independent random variables $X_{1},\ldots ,X_{n}$ $\Pr(X_{1}+\cdots X_{n}>x)\sim \Pr(\max(X_{1},\ldots ,X_{n})>x)\quad {\mbox{as }}x\to \infty .$ This is often known as the principle of the single big jump.

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:

Those that are two-tailed include: 