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. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)

 

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 Lévy distribution;
• the Weibull distribution(威布尔分布) with shape parameter less than 1;
• the Burr[bə:] distribution;
• the Log-gamma distribution.

Those that are two-tailed include:

• The Cauchy distribution, itself a special case of both the stable distribution and the t-distribution;
• The family of stable distributions, excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. Lévy distribution.
• The t-distribution.
• The skew lognormal cascade distribution.

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heavy-tailed distributions

A class of distributions that is often used to capture the characteristics of highly-variable stochastic processes, i.e., more variable than the exponential distribution, is the class of heavy-tailed distributions.

Definition A distribution is heavy-tailed if its complementary cumulative distribution (CCDF), often referred to as the tail, decays slower than exponentially. A typical heavy-tailed distribution is power-tailed .

Definition A distribution has short tail if its CCDF decays exponentially or faster.A typical short-tailed distribution is exponentially-tailed .

 

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Heavy-tailed distributions

Heavy-tailed distributions (also known as power-law distributions) have been observed in many natural phenomena including both physical and sociological phenomena. One example is the geographic distribution of people around the world. Most places in the world are completely empty or barely populated, while there are a relatively small number of geographical locations which are very densely populated.

A distribution is said to have a heavy-tail if:

This means that regardless of the distribution for small values of the random variable, if the asymptotic shape of the distribution is hyperbolic, it is heavy-tailed. The simplest heavy-tailed distribution is the Pareto distribution which is hyperbolic over its entire range .

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