# 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.

## Definition of Heavy-tailed Distribution

The distribution of a random variable *X* with distribution function is said to have a heavy right tail if^{[2]}

This is also written in terms of the tail distribution function as

This is equivalent to the statement that the moment generating function of , , is infinite for all ^{[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 is said to have a long right tail^{[4]} if for all

or equivalently

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 with common distribution function the convolution of with itself, is defined by:

The n-fold convolution is defined in the same way.

A distribution is subexponential^{[5]} if

This implies^{[6]} that, for any ,

The probabilistic interpretation^{[7]} of this is that for a sum of independent random variables

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:

- the Pareto distribution;
- the Log-normal distribution;
- the Weibull distribution;
- 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.

## References

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