Fat-tailed distribution

A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. Different research communities favor one or the other largely for historical reasons, and may have differences in the precise definition of either. Fat-tailed distributions have been empirically encountered in a variety of areas: physics, earth sciences, economics and political science. The class of fat-tailed distributions includes those whose tails decay like a power law, which is a common point of reference in their use in the scientific literature. However, fat-tailed distributions also include other slowly-decaying distributions, such as the log-normal. The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. That is, if the complementary cumulative distribution of a random variable X can be expressed as then the distribution is said to have a fat tail if . For such values the variance and the skewness of the tail are mathematically undefined (a special property of the power-law distribution), and hence larger than any normal or exponential distribution. For values of , the claim of a fat tail is more ambiguous, because in this parameter range, the variance, skewness, and kurtosis can be finite, depending on the precise value of , and thus potentially smaller than a high-variance normal or exponential tail. This ambiguity often leads to disagreements about precisely what is or is not a fat-tailed distribution. For , the moment is infinite, so for every power law distribution, some moments are undefined. Note: here the tilde notation "" refers to the asymptotic equivalence of functions, meaning that their ratio tends to a constant. In other words, asymptotically, the tail of the distribution decays like a power law.