Heavy-tailed distribution

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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.

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

[edit] 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^[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 F, M_F(t), is infinite for all t > 0^[3].

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

[edit] Definition of long-tailed distribution

The distribution of a random variable X with distribution function F 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.

[edit] Subexponential distributions

Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables X₁,X₂ with common distribution function F the convolution of F with itself, F ^{* 2} is defined, using Lebesgue-Stieltjes integration, by:

The n-fold convolution F ^{* n} is defined in the same way. The tail distribution function is defined as .

A distribution F on the positive half-line is subexponential^[5] if

This implies^[6] that, for any ,

The probabilistic interpretation^[7] of this is that, for a sum of n independent random variables with common distribution F,

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

A distribution F on the whole real line is subexponential if the distribution is^[9]. Here 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 ⁺ = max(0,X) is subexponential.

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

[edit] Common heavy-tailed distributions

All commonly used heavy-tailed distributions are subexponential.^[10]

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.

[edit] 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. ^ Willekens, E. Subexponentiality on the real line. Technical Report, K.U. Leuven(1986)
10. ^ Embrechts, Kluppelberg, Mikosch, Modelling Extremal Events, 1997

[edit] See also

• Fat tail
• Leptokurtic
• The Long Tail