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. Roughly speaking, “heavy-tailed” means the distribution decreases more slowly than an exponential distribution, so extreme values are more likely. 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 three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions, and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class, introduced by Jozef Teugels. 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 finite 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 considered to be heavy-tailed. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)
Definitions
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 the moment generating function of X, MX(t), is infinite for all t > 0. That means
∫ − ∞ ∞ e t x d F ( x ) = ∞ for all t > 0. {\displaystyle \int _{-\infty }^{\infty }e^{tx}\,dF(x)=\infty \quad {\mbox{for all }}t>0.}
This is also written in terms of the tail distribution function
F ¯ ( x ) ≡ Pr [ X > x ] {\displaystyle {\overline {F}}(x)\equiv \Pr[X>x]\,}
as
lim x → ∞ e t x F ¯ ( x ) = ∞ for all t > 0. {\displaystyle \lim _{x\to \infty }e^{tx}{\overline {F}}(x)=\infty \quad {\mbox{for all }}t>0.\,}
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 > 0,
lim x → ∞ Pr [ X > x + t ∣ X > x ] = 1 , {\displaystyle \lim _{x\to \infty }\Pr[X>x+t\mid X>x]=1,\,}
or equivalently
F ¯ ( x + t ) ∼ F ¯ ( x ) as x → ∞ . {\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. 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 This section describes the concept of subexponential in the context of heavy tailed distributions. There is an almost opposite definition of subexponential distributions in the context of light-tailed sub-Gaussian distributions. Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables X 1 , X 2 {\displaystyle X_{1},X_{2}} with a common distribution function F {\displaystyle F} , the convolution of F {\displaystyle F} with itself, written F ∗ 2 {\displaystyle F^{*2}} and called the convolution square, is defined using Lebesgue–Stieltjes integration by:
Pr [ X 1 + X 2 ≤ x ] = F ∗ 2 ( x ) = ∫ 0 x F ( x − y ) d F ( y ) , {\displaystyle \Pr[X_{1}+X_{2}\leq x]=F^{*2}(x)=\int _{0}^{x}F(x-y)\,dF(y),}
and the n-fold convolution F ∗ n {\displaystyle F^{*n}} is defined inductively by the rule:
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